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A TREATISE
ON
CONIC SECTIONS
AND THE
APPLICATION OF ALGEBRA TO GEOMETRY.
By J. HYMERS, D.D.
FELLOW AND TUTOR OF ST JOHN’S COLLEGE CAMBRIDGE.
THIRD EDITION, REVISED AND ENLARGED.
CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS,
FOR J. & J. J. DEIGHTON; AND T. STEVENSON, CAMBRIDGE;
RIVINGTONS; AND G. BELL, LONDON,
M.DCCC.XLV.
CONTENTS.
SECTION I.
ON THE METHODS OF DETERMINING THE POSITION OF A POINT
IN A PLANE,
ART. PAGE
1— 9. RecraneutaR and oblique co-ordinates. Polar co-ordinates... 1
10, 11. Equation to a curve. Locus of an equation ..........cecececeees 6
SECTION II.
ON THE STRAIGHT LINE.
12—29. Straight line referred to rectangular co-ordinates .............+ 7
380—35. Straight line referred to oblique co-ordinates ............seseseeee 18
36, 37. Straight line referred to polar co-ordinates ............ssseseseees ya
SECTION III.
38—44. Transformation of co-ordimates ............ssecseeseceescececesseesece 22
SECTION IV.
ON THE CIRCLE.
45—54, Various forms of the equation to the Circle ...............000e0 27
bp—60. Tangent and normal .to a circle..........ccesscesesonesessccsecseceafe 31
SECTION V.
61—68. On the different orders of curves, and on the division of Conic
Sections, or curves of the second order, into three species.. ... 3d
SECTION VI.
ON THE PARABOLA.
69—75. Various forms of the equation to the Parabola ...............6+5 41
76—88. Tangent and normal to the’ parabola..........ccssccsscssseeeceseeene 45
89—103. The parabola referred to oblique co-ordinates............. ive isan Oe
CONTENTS.
SECTION VII.
ON THE ELLIPSE.
ART. PAGE
104—120. Various forms of the equation to the Ellipse .................. 60
121—137. Tangent and normal to the ellipse...........cccccssscssceceeseece 70
138—156. The ellipse referred to its conjugate diameters ............6+. 82
SECTION VIII.
ON THE HYPERBOLA. |
157—173. Various forms of the equation to the Hyperbola ............ 95
174—185. Tangent and quormal to the hyperbola ............ssseeeeeeeeees 103 |
186—204. The hyperbola referred to its conjugate diameters ......... 108 :
205—211. The hyperbola referred to its asymptotes .......:..cccsseseees 117
SECTION IX.
212—221. On the sections of the Cone and Cylinder...........ss000000+- 123
SECTION X.
222—246. On the general equation to Curves of the Second Order, and
on certain general properties of Algebraical Curves ............ 131
247—249. On tracing Curves from their equations.............cs0cseee0- 150
Problems
POOP H HOC E EHH EH AES EEE OOH EEE REELED EHO OTE OES OOOO EEE EE EELS OOEES
SECTION XI.
250—280. On Curves of the third and fourth and higher orders ; and
on the singular points of curve lines
CeCe ee eerecrereeeseceesceeseseees
SrupEnrs reading this work for the first time may confine their attention
to the following Articles:
1—28, 45—48, 55—57, 68-78, 75—82, 104—111, 115, 117—126, 157—162.
Or, they may omit only the following Articles:
i 32—37, 50—54, 58—67, 74, 83—88, 96—103, 116, 132—1387, 150—154,
) 201—204, 211, 215—219, 222—280,
CONIC SECTIONS
AND THE
APPLICATION OF ALGEBRA TO GEOMETRY.
SECTION I.
ON THE METHODS OF DETERMINING THE POSITION OF
A POINT IN A PLANE,
Rectangular and oblique Co-ordinates. Polar Co-ordinates.
1. In order to determine the position of a point in
a plane, some fixed point in the plane is taken for the
origin of co-ordinates; and through it are drawn two fixed
lines, called the co-ordinate axes, at right angles to one
another.
Then if the perpendicular distance (to which the name
ordinate is given) of a point from each of the co-ordinate
axes be assigned, its position will be completely determined.
For let A (fig. 1) be the origin of co-ordinates, X'AX,
Y’ AY, the co-ordinate axes, P any point, and PM, PN, the
perpendiculars let fall from it upon the co-ordinate axes;
these perpendiculars together are called the rectangular co-
ordinates of P, and as their values change for the different
points of the plane, they are denoted by the variables a and y.
Then the point P will be determined in position, if we
know the values of its two co-ordinates; that is, if we
know that for that point v=a, y=6; for if along 4X we
measure AN =a, and through N draw an indefinite line
parallel to AY, this line will contain all points in the
plane whose distance from AY is a, or for which w=a,
and therefore the point in question; similarly, if we measure
along AY the distance AM =b, and through M draw an
indefinite line parallel to AX, this line will contain the
1
2
point in question; therefore these two lines MP, NP, will,
by their intersection in P, determine one single position for
the point whose co-ordinates are wv =a, y=); which position,
as we see, coincides with the angular point opposite the
origin, of the rectangle constructed with the sides AN, AM,
equal to the two given co-ordinates.
2. Instead of the perpendicular PM, its equal AN is
commonly used to determine the position of the point P;
and the two AN, NP, are called the co-ordinates of P, and
are denoted by w and y; the former, for the sake of dis-
tinction, being called the abscissa, as being cut off from LX,
and the latter, which is parallel to the other axis AY, the
ordinate.
When the point is given, and consequently its co-ordi-
nates known, they are usually represented by the first letters
of the alphabet a, 6, &c. as above; or by the accented
letters a’, y’, or a”, y; and the point is called the point
(a, b), the point (a’, y’), &c.; also the axes of the co-ordi-
nates AX, AY, are often called the axis of w and the axis
of y.
3. The determination of the point P will not however
be complete, unless we take into account the signs of the
quantities a, 6, in the equations
Pas “y= b;
in order to measure these distances, when they are positive,
along the positive parts 4X, AY, of the co-ordinate axes;
or along the negative parts 4X”, AY’, of the axes produced
in the contrary direction, when they are negative; as is
explained in Trigonometry, (Art. 20). For since the co-
ordinate axes, which must be supposed to be prolonged inde-
finitely, form about the origin four angular compartments,
there are four positions in which P might be situated at
absolute distances a, b, from the co-ordinate axes; and it
is only by attending to the algebraical signs, with which the
values of those distances are affected, that we shall be en-
abled to select the true position of the point. The direction
of the negative abscissee is quite arbitrary, as is also that
3
of the negative ordinates; we shall however, according to
the usual practice, measure the positive abscisse from the
origin towards the right, and the negative abscissee from the
origin towards the left; and-the positive ordinates we shall
measure upwards from the axis of aw, and the negative ordi-
nates, downwards. Hence if the point P be situated in the
compartment AAY, both its co-ordinates are positive; if in
the opposite compartment X’AY’, both are negative; and for
points in the compartments X’4Y, XAY’, we must have
respectively
e=—a, y=b; w=a, y=—-b;
also for points in the axis of x, and axis of y, we shall
have respectively
V=a,,Y=0; &#=0, y=b;
and for the origin, # = 0, y =0.
4. Sometimes it is requisite to take the co-ordinate axes
not at right angles, but inclined at a given angle to one
another; in which case, the system of co-ordinates is called
oblique.
Thus (fig. 2), if XAX’, YAY’, be two lines drawn
through the point 4, and intersecting one another at a given
angle; and if from any point P in the plane YAY, PM, PN,
be drawn respectively parallel to AX, AY, and meeting those
axes in M and N; PM or its equal AN, and NP, are the
co-ordinates of P referred to the oblique axes AX, AY.
5. ‘lo find the distance of a point from the origin in
terms of its co-ordinates.
Let P be the point (fig. 1), dN=a', NP =y’, its given
co-ordinates.
Join AP, and let 4P =d; then from the triangle ANP,
right-angled at N,
AP’? = AN’ + NP*, or P= a? + y”,
daa! + y”,
6. To find the distance between two points in terms of
their co-ordinates; and the angle of inclination of the line
which joins them, to the axis of a.
1—e
4
Let P’ be a point (fig. 3) whose co-ordinates are # and
y ; and P any other point whose co-ordinates are w and y;
join P’P, and draw P’Q parallel to AX and meeting the or-
dinate of P in Q; then from the triangle PQP’, right-angled
at Q,
PP? = PQ + PQ,
or @ = (#-av)’?+(y-y)’,
d=V(~- a+ ty-y'y.
Both in this formula, and in that of Art. 5, we take
the radical with a positive sign, as the question only relates
to the absolute distance of the points.
Next, let a be the angle which P’P forms with P’Q, and
which is equal to the angle at which, if produced, P’P would
be inclined to the positive part of the axis of a;
PQ. y-y
PQ L—- a
It is important to observe that by the distance P’P is
meant the distance measured from P’ to P, and not from
P to P’; and by the angle which P’P forms with Aa, is
meant the angle which a line AX parallel to P’P through
the origin would form with the axis of #, X being always
on the same side of A that P is of P.
then tana =
7. Suppose the co-ordinates to be oblique, and the axes
of the co-ordinates to be inclined to one another at an angle
w; then, for the distance of a point from the origin, by Tri-
gonometry (Art. 92) we have (fig. 2)
AP*? = AN? + NP*-—24AN.NP cos ANP,
but cos ANP = — cos XAY = — cosw,
._ =e +y’ +2ry cosa;
and for the distance between two points we have, in a
similar manner, from the triangle PQP’ (fig. 3), in which
ZPQP=r-P'QN=7 -PNX=7-2,
D=(x—a)+(y-y)4+2(a@—- 2’) (y-y') cose.
5
8. There is also another mode of determining the posi-
tion of a point in a plane, viz. by means of its distance
from a given point or pole, and the angle which that. dis-
tance makes with a fixed line or axis in the plane.
Let 4A (fig. 4), be the origin or pole, and 4X a fixed
line or axis; and P any point in a plane passing through
AX. Join AP, then AP is called the radius vector, and
is usually denoted by r, and the angle PAX is called the
angle of revolution, and is denoted by @; and r and @ are
called the polar co-ordinates of P; and if given values r = d,
0 =a, be assigned for them, the position of P will be com-
pletely determined.
The angle of revolution may receive any positive value
from zero to infinity ; and it is measured from the initial line
or axis always in the same direction, which, according to the
usual practice, we shall assume to be upwards; and the radius
vector is measured from the pole along the line bounding that
angle, and may have any positive value from zero to infinity.
Sometimes, however, in order to embrace all the branches of a
curve in the same polar equation, it is necessary to admit
negative values of 7, and to measure them from the pole along
the radius vector produced backwards ; also, if negative values
of 0 be admitted, they must be measured from the initial line
downwards.
9. To express the distance of two points from one
another in terms of their polar co-ordinates.
Let P’ be a point (fig. 4) whose polar co-ordinates are
ry and 6’, and P any other point whose co-ordinates are
rand @; then PAP’ =0-6'; and, joining PP’, we get
from the triangle PAP’,
PP’ or d=\/r* +r" — 2rr’ cos (0-0).
Equation to a Curve. Locus of an Equation.
10. As we are able, in the mode explained above, to
determine the position of a point in a plane by means of its
co-ordinates, we may suppose a curved line to be traced ona
6
plane, and each of its points to be referred to two known axes ;
and that we have between the abscissa and ordinate of each
point an invariable relation. In a great many cases, it happens
that this relation is of a nature to be expressed by an equation
between the abscissa and ordinate; and this equation, when
obtained, enables us to find either of those quantities by means
of the other; so that, giving to the abscissa, for instance,
arbitrary values, we can deduce from the equation correspond-
ing values of the ordinate; and we thus determine as many
points of the curve as we please.
The equation which expresses, generally, the invariable
relation of the abscissa and ordinate of every point of a curve
to one another, is called the equation to the curve: and, con-
versely, the curve is called the locus of the equation.
Similarly, the equation which expresses the invariable re-
Jation of the radius vector and angle of revolution of every
point of a curve to one another, is called the polar equation to
the curve.
11. All lines are regular, or irregular; irregular lines,
described, as it is termed, liberé manu, are not subjects of
mathematical investigation, and cannot be represented by
equations; but regular lines, described according to some con-
stant law which determines the position of all their points,
can be represented by equations. This idea of regular lines
agrees with the geometrical loci of the ancients. ‘They gave
that name to those lines of which every point was equally
proper to solve an indeterminate geometrical problem. Thus
a circle was said to be the locus of the vertices of all triangles
on a given base and having a given vertical angle. Des Cartes
first adopted the method of expressing, by an algebraical equa-
tion, the nature of lines. The object of the following Sections
will be to investigate the equations to curves, and from those
equations to discover their geometrical properties by means of
interpretations made according to the laws of Algebra.
SECTION ILI.
ON THE STRAIGHT LINE.
Straight Line referred to Rectangular Co-ordinates.
12. We will now suppose the locus of the point P to be
a straight line, as defined in Geometry; and proceed from
some of the fundamental properties of a straight line to deduce
its equation; that is, an equation expressing an invariable
relation which is satisfied by the co-ordinates of every point
in it.
13. To find the equation to a straight line.
Let A be the origin (fig. 5), AX the axis of x, AY that
of y; RT the given straight line meeting these axes in 7' and
B respectively, "P any a in it, and NGS 2, PN =y the
co-ordinates of P.
Let AB =e, and the tangent of angle PTN =m. Draw
BQ parallel to 4X, meeting PN in Q; then by Trigonometry
(Art. 90)
PQ = BQ.tan PBQ = AN.tanPTN = mz,
and PN = PQ +QN = PQ + AB,
~Y=MPt+e;
and as this relation is satisfied by the co-ordinates of every
point in the line, it is the equation required.
Ozs. The meanings of the constants m and ¢ are to
be particularly noticed ; c is the part of the axis of y inter-
cepted between the straight line and the origin, or the
ordinate through the origin ; m is the tangent of the angle
which that part of the line which falls above the axis of 2,
8
makes with the axis of # produced in the positive direction.
They remain the same for the same line, but are different
for different lines, and are called arbitrary constants, or
parameters ; in general every straight line has two of them,
and therefore a straight line may be drawn fulfilling two
conditions.
14. The equation y=mw +c, which is the most convenient
form and .the one commonly employed, represents a straight
line when determined by the conditions of passing through
a known point in the axis of y, and making a given angle
with a fixed line, viz. the axis of x; so that m is a number
or ratio, denoting the tangent of the angle; and c denotes
a line, viz. the distance from the origin, of the point in the
axis of y through which the line passes.
If c=0, the line passes through the origin, and its
equation is y= ma; also if m=0, the line is parallel to
the axis of wv, and its equation is y=c. Similarly, the
equation # = a, since it belongs to all points whose distance
from the axis of y is a, represents an indefinite line parallel
to that axis; and the equations y= 0, # = 0, represent the
axis of wv, and the axis of y, repectively, (Arts. 1 and 3).
“15. The equation to a straight line may also be put
under the two following forms, which are sometimes useful.
Let BT' (fig. 7) be the line intersecting the positive
parts of both the co-ordinate axes, and let AB=c, tan BT X=m,
as before; and AJ7'=a; then the equation to the line is
YyY=MeL+C3
but m = tan BT'X = —tanBTA=--,
the equation to a straight line when determined by means
of the portions of the positive co-ordinate axes intercepted
between it and the origin.
S
Also, if a perpendicular upon the line from the origin,
AD =p, and 2DAX =a, we have, by Trig. (Art. 23),
m= —tan BT'A = — cota, and c = ae (Trig. 98),
sina
es _aecota +——; or y sina + &cOSa = p,
sina
the equation to a straight line when determined by the per-
pendicular upon it from the origin, and the angle which the
perpendicular makes with the axis of x produced in the positive
direction.
16. The indeterminate equation of the first degree be-
tween two variables, is, in its most general form,
Av+ By+C=0,
which in all cases is the equation to a straight line. For
by putting
C
Sea LS pat ees,
B
we reduce it to the form y—mxv—c =0, or y=ML+0,
which coincides with the equation to a straight line.
17. OF. Mm = F
1+tana.tangd 1+mt
—t
therefore y —y' = -(w—- 2’) is the required equation.
1+mt
Similarly if PT” be the other line answering the con-
ditions of the Problem and z PT" X =a" then a’ =a+ d,
m+t 1 : d fe . , a /
m= and the required equation is y—y = U-~ax').
~ 1[-—mt ‘d ‘4 I~9 1—mt )
16
If the lines are to be parallel, ¢=0, and as before the
equation is
y-y =m(#—a’).
If the lines are to be perpendicular to one another, ¢= 0,
and therefore the equation (since m in the numerator, and
1 in the denominator, vanish with respect to ¢) becomes
’ 16. , ’ 1 ,
Yr = — (0 — &# or Y¥ — = ~——(V— @#).
Yad ater ), ory-y oA )
27. Having given the co-ordinates of a point, and the
equation to a straight line; to find the length of the per-
pendicular dropped from the point upon the line.
Let a’, y’, be the co-ordinates of the given point, and
y=ma-+ec the equation to the given line; then the equation
to the perpendicular will be
1 ,
Pf ati ae Ui
In order to get the co-ordinates of the point of intersection
of the given line and the perpendicular, we must, as in Art. 24,
deduce from their equations values of # and y; to make the
process easier, put the first, y = mw +c, under the form
y—y =m(@—a)+co%+ma' -y ;
combining this with the equation to the perpendicular, and
taking for the unknown quantities, the differences y—y’,
x —a, we get
,_m(y! —ma' -c)
’ r
TE Sa) VALI (bs
et—-t= ees °
/
—_y=
1 +m? : J 1 +m?
>
values from which it is easy to deduce the co-ordinates a, and
y, of the foot of the perpendicular. But if we denote by p
the length of the perpendicular intercepted between the point |
and the given line, we have (Art. 6), |
p=/ («@-a#P+y-yy);
17
therefore, putting for w—w and y—y’ their values,
y —mx —Cc
1 Ps = eg
Yfl+m
As the value of p must be positive, we must take the
upper or lower sign, according as the numerator y —Me —c
is positive or negative.
The double sign may be explained, as having reference
to the face of the straight line upon which the perpen-
dicular falls. For suppose the line to revolve about B
from the present position, in which the perpendicular is
positive; then as it moves up to P, the perpendicular di-
minishes, and vanishes ; and when it passes P, the perpen-
dicular, falling upon a different face, becomes negative, and
continues so till the line, after half a revolution, returns to
its first position; the line has then the same equation as
before, but has a different face turned towards P; and the
perpendicular has the same value as at first, but with a
contrary sign (fig. 9).
If the given point is situated in the origin, v= 0,
y = 0, and the value of p is reduced to
Cc
J 1 +m
28. The result of the preceding Article may be readily
obtained as follows.
p=+
Let y=mx+ce be the equation to the given line MT
(fig. 9), and AN = x, PN=y’, the co-ordinates of the point
P; then the perpendicular
PQ = PReosRPQ = PRoos RTN.
But PN=y’, and RN=ma' +c, «. PR=y' -—ma'-c;
1 1 ‘
“/i1 tant RENO A/D
/ ’
y —-me —C
Garaen
also cos RT'N =
ete perpendicular p=
18
29. Hence also, if through a given point a line be
drawn cutting a given line at a known angle, we can find
the distance of the given point from the point of inter-
section of the lines. For if PS' (fig. 9) be a line passing
through the point P, and cutting the line BM at an angle
PSM = a, by drawing PQ perpendicular to BM, we have
, ,
y —mu —c y —ma —c
= Sine SP = 5 ee .
/ 1 +m? sina 4/1 +m
SP sina=QP =
Straight Line referred to Oblique Co-ordinates.
350. To find the equation to a straight line referred
to oblique co-ordinates.
Let the axes of the co-ordinates be inclined to one
another at an angle w, and suppose the line PT’ (fig. 10),
to cut the axis of w at an angle a. Let AN=a2, NP = Ys
be co-ordinates of any point P, and draw BQ parallel to AX,
meeting the ordinate of P in Q;
PQ sin PBQ sina “sina
then —— = = , or PQ=
BQ sin BPO N No Dayo ie
sin (w — a)”
and NQ = AB =c:; therefore the required equation is
Hence if a straight line ‘referred to oblique axes, be
represented by the equation y=ma-+ce; m, the coefficient
of xv, expresses the ratio of the ‘sines of the angles which
the line makes respectively with the axes of # and y; and ¢,
as before, is the ordinate through the origin.
& sila
In using the equation y = + C, we must re-
sin (w — a)
member, with respect to the constants Involved, (1) that
c is a positive or negative quantity, according as the line
cuts the axis of y above or below the origin; (2) that
w is the angle VAX formed by the positive parts of the
19
co-ordinate axes, and not the adjacent angle YAX': and.
(3) that a is the angle PTX formed by the portion of the
line which is situated above the axis of x, with the positive
part of that axis.
31. It is easily seen that when a line is determined
by the portions of the co-ordinate axes intercepted between
it and the origin, its equation is of precisely the same form
as when the co-ordinates are rectangular; for let PT’ be
the line (fig. 12), AP=c, AT’ =a; and let and y be the
co-ordinates of any point Q; we get from the similar triangles
MPT QNT,
APN c y ev Y
Te ENT OE at Ne in? eee
Also when a line is determined by the condition of passing
through two given points, or of being parallel to a given
line, its equation is of the same form whether the co-ordinates
be rectangular or oblique; in the following cases, the results
are different.
32. To find the angle between two lines whose equa-
tions are given, referred to oblique axes.
First, to calculate the angle which a line whose equa-
tion is given, makes with the axis of 2, Let y= mev+e
be its equation, and w the angle of inclination of the axes ;
and let the line make an angle a with the axis of «w, and
therefore an angle w —a with the axis of y, then
sIn a
—__———_ = M,
sin (w — a)
: : Mm sin w
which gives tana = :
1 + m COS w
Next, let y = mx +e be the equation to another line
referred to the same axes, making an angle a’\ with the
axis of @;3
m’ sin w
., tana’ =
1 +m! cosw
20
Let @ be the angle between the lines,
He tana’ — tana
HP then tang = tan(a’ — a) = 5
i 1+tana.tana
| (m" — m) sin w
iy 1+ mm! + (m+ m’) COS w
HN | 33. Hence if d = dar, then
ih 1+mm +(m4+m’) cosy =0;
i} ; 1 +mcosw
3
. M + COS w
Hi the condition in order that two lines referred to oblique
i axes, may be perpendicular to one another. The condition
Is of parallelism is the same as for rectangular co-ordinates.
34. To find the perpendicular distance of a given point
ih from a given line, referred to oblique axes,
Let wv’, y’, be the co-ordinates of the given point Q
(fig. 11), and y= ma +c the equation to the given line CN
which makes an angle a with the axis of vw; then
sina = m sin (w — a);
|
also let QP be the perpendicular let fall from Q upon CN;
then
sin a
i | QP = QN.sin (w — a) = (y' — ma’ —c)
]
4 m
! Mm sin w ; tana
j butitan’ay= ————— an sing = ee
Mt rik + ™ COS w /1 + tan? a
{ Mm sin w Dye
/1+2m cosw +m
(y — ma’ —¢) sinw
AY x
/1+2m cosw +m?
35. To find the area of a trapezium.
Take one of the sides of the trapezium P'N’NP (fig. 3)
for the axis of x, and a line parallel to its two parallel sides
21
for that of y; and let w, y be co-ordinates of P, and wv’, y’
those of P’.
Then area of P'N’NP = area of parallelogram P’N
+ area of triangle P’PQ
=(#-2')y sinw + 4(@—- a’) (y-y) sin
=1L(r-w) (y +y) sinw.
Straight Line referred to Polar Co-ordinates.
36. To find the polar equation to a straight line.
Let A be the pole (fig. 12), XA the initial line, PT
the proposed straight line, and 4P=7, 2XAP=0 the
polar co-ordinates of any point P in it; AQ =p a perpen-
dicular upon it from the pole, and zXAQ=a the angle
which that perpendicular makes with the initial line ;
then AP = AQ sec QAP,
or r = psec (0 — a).
If the proposed line be perpendicular to the initial line so
that a=0, the equation becomes += psecQ@. And if we
choose to determine the line by the angle, and by the
distance from the pole, at which it cuts the initial line, so
that AT =a, TENA stl HAND aN aca
, Gant 8 Has taeR a sin( — 6)
37. Since the polar equation to a straight line becomes
rcos(@—a) =p, or rcos@.cosa+7sin@.sina=p;
it appears that every equation of the form
Arcos@ + Brsin@ + C =0,
is the polar equation to a straight line.
S ECRTON IT
ON THE TRANSFORMATION OF CO-ORDINATES.
38. As the equation to a curve does not remain the
same when we change the axes of the co-ordinates to which
it is referred; it is of great importance, in investigating
the form and properties of a curve from its equation, to
give the origin of the co-ordinates such a position, and
the axes such directions and inclination, as will allow the
equation to the curve to appear under the simplest form
possible.
Moreover it is frequently required, when we know the
equation to a curve referred to an assumed system of axes,
to find the equation which represents the same curve when
referred to new axes, whose positions are given with respect
to the former.
This will be effected, when we know, for any point of
the curve, the values of the old co-ordinates in terms of
the new ones; for then, by substituting these values in the
proposed equation, we shall obtain a relation between the
new co-ordinates, which is true for every point of the curve
under consideration. In this consists the transformation of
co-ordinates. ‘The problem therefore to be resolved is, to
express the primitive co-ordinates of any point in terms of
the new co-ordinates and of the known quantities which fix
the position of the new origin and axes; which we shall
now consider in the following separate cases.
It is evident that if in any equation we write — # for x,
the effect will be to reverse the positive direction of the
abscissee ; and similarly, for the ordinates.
39. ‘To change the origin of co-ordinates without
altering the direction of the axes.
Let A (fig. 15) be the origin of co-ordinates, and AN = 2,
PN =y, the co-ordinates of any point P; AM =h, MA =k,
23
the co-ordinates of the new origin 4’, and 4’N’ =a’, N’P=y,
the co-ordinates of the same point P referred to the new
axes A’X’, A’Y’, parallel to the former; then
w=AM+ AN =h+ao’, and y= AM+NPH=kK+y;,
which are values of the primitive co-ordinates in terms of
the new co-ordinates; and if these values be substituted
for w and y, we shall have the equation to the curve
referred to the origin A’, the axes retaining their former
directions.
40. To change the directions of the axes without altering
the origin, supposing both systems to be rectangular.
Let XAX’ = YAY’ =6 (fig. 16) be the angle at which
the new axes AX’, AY’, are inclined to the original axes
mA. AY.
Let AN=a, PN=y; AN =2', PN’ =y’'; be the co-
ordinates of the same point P referred to these respective
axes.
Draw N’Q, N’M, parallel and perpendicular to AX,
then « = AM — QN’ =2' cosO—-/y’ sin 8,
y= MN’ + PQ = «'sind+y' cos,
which are values of the primitive co-ordinates in terms of the
new co-ordinates; and if these values be substituted for w and
y, we shall have the equation to the curve referred to the
axes AX’, AY’.
41. To change the direction of the axes without altering
the origin, supposing both systems to be oblique.
Let AX, AY, (fig. 17) be the primitive axes inclined
to one another at an angle XAY=wo, and AX’, AY’, the
new axes determined by the angles X’AX =a, YAX= ®B,
which they respectively form with the primitive axis of w.
Let P be any point, 4AN=a2, NP=y, its primitive
co-ordinates, and AN’=.2, N’P=y', its new co-ordinates.
24
Then, denoting by wy the angle XAY contained by the
axes of w and y produced in the positive directions, and
similarly of the others, and drawing PQ, PR, perpendicular
to AX, AY, and N’S, N’T, parallel to those lines, we
have
Thea ele:
or #sinwy =a sina’y +y'siny’y;
, Sin (w — a) Fe , Sin (w — B)
sin @ sin a)
Also QP = QS + SP,
or ysinay=“ sinw ae +y'siny' a;
sma ,sinf
ye) YS egy ae ;
sin w sin w
and if these values of the primitive co-ordinates, which are
expressed in terms of the new co-ordinates and known quan-
tities, be substituted for # and y in the equation to a curve,
we shall obtain the equation to the curve referred to the
new axes.
Ozss. In making use of these formule, we must recollect
that w represents the angle AY contained by those portions
of the primitive axes along which the positive co-ordinates
are measured, and varies from zero to 7; and that a and B
denote the angles formed by AX’, AY’, the positive direc-
tions of the new axes, with the positive part AX of the
primitive axis of w; and that each of these may receive
any value from zero to 27.
42. The. formule of the preceding Art. are not often
employed in the above general state; and we may deduce
from them, as particular cases, the following, the use of
which is more frequent.
° . T T
First, by making ater. and Rie sHa, we fall upon
25
the formule of the second case for passing from one system
of rectangular co-ordinates to another system also rectan-
gular.
Secondly, by making w =
, we obtain the formule for
0] 9
passing from a rectangular to an oblique system of co-ordi-
nates, which are
, (
v=2 cosa+y' cos (3,
o Le e
y=usinat+y sinf.
Thirdly, by making B=> +a, we obtain formule for
passing from an oblique to a rectangular system of co-
ordinates, which are
w sin(w—a) y' cos(w —a)
€ —S=
sIn w sIn w
, e
xv sina y cosa
sIn w sIn w
43. In all the preceding transformations except the first,
we have supposed the origin to remain unaltered; if, how-
ever, the origin is to be changed as well as the direction
of the axes, we must employ the formule
waa th, y=y +h,
where h, k are the co-ordinates of the new origin parallel to
the primitive axes, and a”, y”, denote the values of w and
y, found in each of the preceding cases.
44, To transform rectangular into polar co-ordinates,
and conversely.
Let AN=2, PN=y, (fig. 4) be the rectangular co-
ordinates of any point P in a curve referred to the rect-
angular axes 4X, AY; and let d4P=r, PAX =0, be the
polar co-ordinates of P.
26
Then AN = AP cos PAN, or v« =r cos 0,
PN=APsin PAN, or y=rsin@;
and any equation between w and y may be transformed into
the polar equation, by substituting these values for # and y.
; arte Y _1 JY
Also, since 7 = \/a? + y’, and tan9==— or @=tan7!-,
wv shi
any equation between 7 and @ may be transformed to rect-
angular co-ordinates, by substituting these values of x and @.
If the pole do not coincide with the origin of rectan-
gular co-ordinates, then if h and & be the co-ordinates of the
pole, the quantities to be substituted for # and y, in order to
get the polar equation, will be
ev=h+rcos0, y=k+rsin0.
Or, if we wish at the same time to take for the initial line,
not a parallel to the axis of # but a line inclined to it at
an angle a, the substitutions will be
vw=h-+rcos (0+ a), y=k-+rsin (0 + a).
SECTION IV.
ON THE CIRCLE.
Equation to a Circle under various Forms.
45. To find the equation to a circle when referred to
two diameters at right angles to one another as axes.
Let P be any point in the circumference (fig. 18), CN = a,
NP =y, its co-ordinates, and CP =c, the radius; then from
the right-angled triangle CNP,
CON? NEP O=CE:,
or 7 +y = c,
which is true for every point in the circumference, and is,
therefore, the required equation ; it expresses that the distance
of every point in the curve from the origin, is equal to c.
46. To find the equation to a circle when referred to
any rectangular axes.
Let C be the center, and P any point in the circumference
(fig. 19); draw CB, PN, perpendicular to LX, and CM paral-
lel to AX, and let the co-ordinates of C be AB =a, BC =);
the co-ordinates of P, AN=x2, NP=y; and the radius CP=c.
Then from the right-angled triangle CPM,
CM" MP2 =. GP*.
But CM=BN=a-a, MP=PN-CB=y-5);
therefore we get for the general equation to the circle referred
to rectangular axes, |
(v—a)’+(y -— by =e’.
28
47. From this we may deduce several particular forms
of the equation to the circle, which are worthy of notice.
First, putting a=0, b=0, we have the origin in the
centre, and fall upon the equation already found,
e+ y? = ¢,
Secondly, putting @=c, b= 0, we have the origin at the
extremity of a diameter, and that diameter the axis of w, and
we get
(w- of syne,
or, reducing, oy = 200 — a",
Thirdly, putting 6=0 simply, the centre will be in the
axis of x, but the origin will not be in the circumference;
similarly, putting a =0, the centre will be in the axis of y;
and in these two positions, the equations will be, respectively,
(v7 - a)’ +y? =e’,
w+ (y—b)y=c*.
48. ‘The above general equation to the circle (Art. 46),
when developed, assumes the form
x+y? — 2a” —2by + (a? +? — cc’) =0,
or wv +y°+ Aa+ By+C=0,
which does not contain the product of the variables w and Y>
and in which the coefficient of each of the squares of x and Y>
is unity. Whenever, therefore, an equation of the second
order between rectangular co-ordinates, is such (or by dividing
by the coefficient of «* can be made such) that these con-
ditions are satisfied, the equation cannot represent any other
curve except a circle. In fact, by completing the squares,
we get
(e+ $4) + (y + $B) = 44? + TB -C,
which evidently represents a circle, the co-ordinates of whose
centre are —4A, — +B; and of which the radius
= 1 42 1 Re
=4/ 1 dea
The equation, however, will not in reality represent a circle,
29
unless the quantity 14° + 1B —C is positive; if this quan-
tity is zero, the circle is reduced to a point, namely the centre;
if it is negative, the equation is impossible.
Peculiarities of this sort are offered by the equations
v+y’ —8y —128 + 52=0,
“ty —4y+ 20+ 9=0,
which may be, respectively, reduced to the forms
(vw - 6)? +(y—4)?=0, (w+1)P? +-2)?=—-4.
But the equation a + y’+ 4y — 4e —8=0, by complet-
ing the squares, becomes (# — 2)? + (y + 2)? =16, which re-
presents a circle whose radius is 4, and the co-ordinates of
the centre 2 and — 2.
49. To find the equation to a circle when referred to
oblique axes.
Let, as. before, AB=a, BC=b; AN=a, NP=y;
(fig. 20) be the co-ordinates of the center, and of any point
in the circumference of a circle, referred to the oblique axes
AX, AY, which form with one another an angle w. Draw
CM parallel to 4X; then from the triangle CPM in which
ZCMP = 7 —w, and whose sides are respectively equal to
AN — AB or w—a,\PN—BC or y—5, and CP =c the
radius, we get
(w — a)? + (y—b)? +2(@- a) (y - 6) cosw =e’,
the required equation.
50. The above equation, when developed, becomes
a? + y? + 2ay cosw — 2(a + bcos w) # — 2(b + acosw)y
+a? + b+ 2abcosw —c’ =0,
which is of the form
erry +2rycoswt+ Av+ By+C=0.
Whenever, therefore, an equation of the second order be-
tween oblique co-ordinates of known inclination, is such (or
by dividing by the coefficient of 2° can be made such) that
30
the two squares # and y” have unity for coefficient, and the
rectangle wy has for coefficient twice the cosine of the angle
between the axes, the equation will in general represent a
circle; and the co-ordinates of its centre, a and b, and the
radius ¢, may be determined by the equations a + bcosw=
b+acosw=—43B, a ++ 2abcosw-c =C.
Also, dropping the perpendiculars CD, CE, (fig. 20) upon
the axes, we get
AD=AB+ BD =a4+4 bcos» =
AK =b+ acosw=-4B.
If therefore we take the distances 4D, AE, equal, respec-
tively, to half the coefficients of # and y with contrary signs,
and erect the perpendiculars CD, CE, we shall by the in-
tersection of the perpendiculars determine the position of the
centre C.
Thus, in order that the equation
“+ eyty’ —24a4e—-2ay+a’=0
may represent a circle, we must have 2 cosw=1, or w= dg
and if «, y’, be the co-ordinates of its centre, and c¢ its
radius,
w +y' cosw=a, y' +2’ cosw=a, w+ y%422'y' cosw—c
L
2a
, and c= ta / 3.
2
o
51. To find the polar equation to a circle.
Taking the pole for the -origin, let a, b, be co-ordinates of
the centre of the circle (fig. 21), and ¢ its radius; and let
AP=r, £XAP=6, be the polar co-ordinates of any point
P, supposing the initial line to coincide with the axis of x.
Then, substituting rcos@ for xv, and rsin@ for y, In the
equation
(wv — a)’ + (y — b)? =c’, and expanding, we get
x cos’ @ —2arcos§ + a? + sin? @ — 2br sin 8 + b? = ‘eye
or 7 — 2(acos@ + bsinO)r + a? +B? — c* =0,
(since cos’ @ + sin?@=1) the required equation.
31
This equation will give two values of 7, AP, AP’;
~. AP.AP' = + 0° —c’, which is invariable.
52. If we suppose the pole to be in the circumference,
and the initial line to be a diameter, we have b=0, a=e,
and the equation becomes
r? — 2crcos@ = 0,
*.. r= 2c cos 0;
at which we may arrive immediately by joming AP, PB,
(fig. 18); for the right-angled triangle BAP gives
AP = AB cos BAP, or vr = 2c cos @.
53. Let PY be a tangent to the circle at P (fig. 21),
and AY a perpendicular upon it from the pole; and sup-
pose AY=p, AC =d; then since Z YPC = 90°,
co —
P =sin APY = cos APC = a , (Trig. Art. 93),
Us 27re
or 2cp=7° +c? —d’, a relation between 7 and p.
If the pole be in the circumference, or c = d, this becomes
yo = 2ep.
54. From the preceding equations to the circle, which
assume no other property of a circle than that it is the locus
of a point which is always at the same distance from a
given fixed point, all the theorems relative to the circle
established in geometry, may readily be deduced. We shall
however confine our attention to those which relate to the
tangent.
Tangent and Normal to a Circle.
55. To find the equation to a straight line which shall
touch a circle at a proposed point.
In geometry a line is said to touch a circle when it
32
has only one point in common with the circumference; if
therefore through the two points P, P’ (fig. 22), we draw a
secant PP’, and then make it turn about P, till P’ coincides
with P, the secant in its ultimate position will become a
tangent at P, for it will have only one point in common
with the circumference. ‘This consideration furnishes an easy
method of determining the tangent at a given point of the
circumference.
Let a’, y’, be the co-ordinates of the given point P, and
m the tangent of the angle which the touching line makes
with the axis of w; then its equation will be
y¥—-y =m(a—-a’) (Art. 20),
where m is to be determined.
Let x”, y”, be the co-ordinates of another point P’ in
the circle near the given point, and let a’ be the angle
which the line joining them makes with the axis of x; then
(Art. 6),
Is Ms , / ”
y > —-y* ev +a” oe +a
,
Uae,
? ” 12 TR Lie Y Bato Ta, 7?
LY —w@ w-—-@ > yry y+y
tana =
since, the points being in the circumference, their co-ordinates
must satisfy the equation to the circle; and therefore
m2 "2
§ , Is ‘ ¢ :
yPa=e— a, y= — a, and y? — oy? = a’? — a...
Now let 2” = a’, and y” =y’, so that P’ coincides with Vie
and the secant PT" assumes the position of the tangent PT';
therefore, denoting by a the angle which the tangent forms
with the axis of w, we get
t Xv d , ae ;
m= ana=-—-, an os =-—(t¢-@
ie y-y ; )
for the equation ‘to the tangent; or, multiplying by y’, and
observing that wv + y’ = c, the equation, in its most simple
form, becomes
9
4 ,
yy + aa’ =’,
in which a’, y’, are the co-ordinates of the point of: contact,
and #, y, those of any point in the tangent line.
'
56. The equation to CP is y = Ls x, which compared with
xv
the above equation to P7', shews that CP and PT are at
right angles to one another (Art. 23); that is, the tangent
to a circle at any point, and the radius drawn to the point
of contact, are perpendicular to one another.
Also the equation to the tangent in terms of its in-
clination to the axis of 2, is
C
Pere cena raies ae} +m’, since c* = y(1 +m’);
the lower sign referring to the point P”.
57. To find the equation to the normal at any point
of a circle.
A line PG (fig. 22) drawn through the point of contact
perpendicular to the tangent, is called a normal. The co-
ordinates of the point being a’, y’, the equation to the normal
will be of the form
y—y =m'(e4— 2’), (Art. 20);
and the condition of being perpendicular to the tangent gives
,
1
m=-—=- (Art. 24);
-y-y= a (vw — 2’), or, reducing, y = ai
@ 2
which is the equation to a line passing through the origin,
in this case the centre. Hence all normals to a circle pass
through the centre.
58. To find the locus of the middle points of a system
of parallel chords.
Let all the chords make an angle a’ with axis of x,
and let PP’ be one of them (fig, 22) and «, y, the co-
3
—=-
SS SS
See
=
a Fe
34
ordinates of its middle point V; then proceeding as in Art. 55,
and using the same notation,
, ”
Qy=yY +Y >, 2a u' tu’;
and consequently the locus of V is a line through the centre
perpendicular to the chords.
59. To find the equation to a straight line which shall
touch a circle, and pass through a given point without the
circle,
Let h, &, be the co-ordinates of the given point ; and w’,
y', those of the point of contact, which are unknown ; when
they are found, we shall have, for the equation to the tangent,
t ’ 2.
YY +axv =C';
and as the tangent passes through the given point, its equa-
tion must be satisfied by the co-ordinates of that point,
i KY REE cs
and since the point of contact is in the circumference,
y? +a” =e’,
which are the two equations that serve to determine wv! and 1’.
It is evident that # and y’ will each have two values;
therefore there will be two points of contact; and the equa-
tion
yi+radh=e’,
since it is satisfied by the co-ordinates of the points, will
be the equation (regarding a! and y’ as the variable co-ordi-
nates), to the chord joining the points of contact of two
tangents drawn from the point (h, k); for if an equation
of the first degree between two variables be satisfied by the
co-ordinates of two points, it must be the equation to the
straight line passing through those points.
35
60. When a problem, as in the present case, leads to
two equations between the co-ordinates # and y’ of an un-
known point, each of the equations, taken separately, gives
a geometrical locus in which the point is placed; conse-
quently, if we construct the two loci, we shall have two lines,
the Aintersections of which will determine the points which
satisfy the problem.
The locus of the second equation is the proposed circle ;
the locus of the first is a straight line AB (fig. 23), which is
; Cc c he i:
constructed by taking CA wa CB = Re? and joining AB;
L
then the points 7’, 7”, in which this line cuts the circle are
the points required.
Hither of the equations may be replaced by another which
results from combining them in any manner; and in solving
problems in this way, we must always select the combinations
whose loci are easiest to construct.
Thus, if we subtract the above equations, we get
y?—yk + @?—awh=0, or (y —dk) +(e — Shy =1k + 1h’,
which represents a circle whose centre is O, the middle point
of CP, and radius CO, P being the point through which the
tangents are to be drawn; if then we join CP and bisect it
in O (fig. 23), and with centre O and radius OC describe a
circle cutting the former in 7’, 7", these are the points of
contact, and are determined by a simpler construction than
the former one.
It is evident that the two tangents PT’, PT”, subtend
equal angles at the centre.
2
The value AC =o, which determines the point 4 in
which 7'7” meets the axis of a, is independent of the
ordinate & of P; therefore A will remain in the same posi-
tion, for all positions of P in the indefinite line PM parallel
to the axis of y. If therefore from the several points of
3—2
any straight line (since the direction of the axis of y is
arbitrary), we draw pairs of tangents to a circle and join
the corresponding points of contact, all the secants will
intersect in the same point; and conversely if through any
point we draw different chords and apply two tangents at
the extremities of each, the locus of the intersection of each
pair of tangents will be a straight line.
SECTION V.
ON THE DIFFERENT ORDERS OF CURVES; AND ON THE DIVI-
SION OF CONIC SECTIONS, OR CURVES OF THE SECOND
ORDER, INTO THREE SPECIES.
61. Lines are divided into orders according to the
degree of their equations, the degree being determined by
the sum of the indices of # and y in that term of the
equation (which is supposed to contain no fractional or
Irrational term) where it is greatest.
The straight line is the line of the first order, being
the locus of the equation of the first degree between two
variables; the circle is a line of the second order, or curve
of the second order (these terms being used indifferently),
because its equation is of the second degree.
62. Curves of the second order are those whose equa-
tions involve the squares, or the simple product of the
variables w and y; but no powers or products of them which
are of higher dimensions. Hence the equation to curves of
the second order under its most general form, or, which is
the same thing, the general equation of the second order
between two variables, is
ay’ +bey+ca°+dy+exr+f=0,
which (as will be hereafter shewn) by giving a proper posi-
tion and direction to the origin and axes of the co-ordinates,
can always be reduced to one of the forms
Ay’ + Ba’ =C,
y= Ax,
representing two distinct families of curves; the former those
which have a centre, the latter those which have not a centre.
ee eee
=
- = —_
=
SS eS
38
63. The centre of a curve is a point such that all
lines drawn through it and meeting the curve both ways,
are bisected in it.
An axis of a curve is a line with respect to which the
curve is symmetrically situated.
64. Of the curves represented by the equation
Ay’ + Be=C,
the origin is the centre, and the axes of the co-ordinates
are axes.
For suppose P (fig. 24) to be a point in the curve
having known co-ordinates 2’, y’; then the equation is satis-
fied by these values; and since it contains only even powers
of # and y, it is also satisfied by the same values taken
negatively ; but if we produce PC to P’ and make CP’ = CP,
the co-ordinates of P’ are —a’ and —y’; therefore P’ is a
point in the curve, and PP’ is a chord, and it is bisected
in C; that is, every chord is bisected in C, and therefore
C is the centre of the curve.
Also the curve is situated symmetrically with respect
to the co-ordinate axes; for if in the equation we put
«=CN=2", we get for y two equal values with contrary
signs, PN, P,N; so that for every point situated above
the axis of x, there will be a corresponding point situated
at an equal distance below that axis. Similarly, for a given
value CM =y' of the ordinate, the equation furnishes two
equal values with contrary signs, MP, MP”, of the abscissa.
Hence each of the co-ordinate axes bisects its ordinates at
right angles; and the curve is situated symmetrically with re-
spect to them, or they are axes of the curve.
65. In the equation to curves of the second order that
have a centre,
Ay’ + Be=+C,
having taken care to make the second member positive,
since the coefficients of the variables cannot be both nega-
39
tive together, we can have only two varieties of form; one
with both coefficients positive, the other with one coefficient
negative; so that the equation may assume the two forms
ae: = 1 a he ir.
OO as Cars Os
The curves represented by these are called respectively,
the Hllipse, and the Hyperbola.
In the particular case of C = 0, the equation may assume
ey ; ;
either the form — + - = 0, which can only be satisfied by
fir ie
the values v = 0, y = 0, representing a point, viz. the origin ;
a ¥ Oo y\ (ay
or the form — —-— = (: ae ‘) (= ~ 2) = 0 representing two
Gb: LANGE AG P 3
straight lines; for the equation will evidently be satisfied
by the co-ordinates of any point in either of the lines
66. Of the curve represented by the equation
y = Aa;
the origin is not the centre, since the equation does not
remain unaltered when w and y are changed into —# and —y;
and we shall see hereafter that it cannot have a centre;
also, since only an even power of y enters into the equation,
the axis of w is an axis of the curve, but the axis of y
is not an axis of the curve.
The equation may always be reduced to the form y?=4a.2,
where a@ is a positive quantity; because if A be negative,
we have only to change & into —a, the effect of which will
be merely to reverse the position of the curve. Hence the
second division of lines of the second order offers only one
variety, which is called the Parabola. If A = 0, the equa-
tion becomes y?=0, representing a straight line, viz. the
axis of a.
These three species of curves, to one or other of which all
lines of the second order belong, are called Conic Sections.
ST ee “ ee ee anpmemenrtnanAamagabaii Fi
40
67. Instead of entering upon the discussion of the
general equation of the second order (which may more con-
veniently be reserved for a more advanced part of the work),
we shall now separately investigate the equation to each of
the Conic Sections from a simple definition which embraces all
m of them; and thence determine their figures and properties.
68. Derr. The locus of a point whose distances from
a given fixed point and a straight line given in position,
are always to one another in a constant ratio, is called a
Conic Section.
Thus let § (fig. 25) be the given fixed point, and KX
the line given in position, P a point such that joining SP
and drawing PM perpendicular to KX, the distances SP,
PM, are always to one another in an invariable ratio, then
the locus of P is a Conic Section.
The point S is called the focus, and the line AX the .
directrix. Since SP may be always equal to PM, or always
less than PM in a constant ratio, or always greater than PM
in a constant ratio, there will be a distinct species of Conic
Section corresponding to each of these cases; in the first
case the locus of P is called the Parabola, in the second the
Ellipse, and in the third the Hyperbola.
The condition of SP being equal to, or less than PM,
can only be satisfied when P_ falls on the same side of
KX with §; but that of SP being greater than PM,
may evidently be fulfilled, on whichever side of AX, P
is taken. Therefore the Parabola and Ellipse will lie entirely
on the same side of the directrix, as the focus does, by
which they are described; but the Hyperbola will lie on
both sides of the directrix.
SECTION VI.
ON THE PARABOLA.
Various Forms of the Equation to the Parabola.
69. To find the equation to the parabola.
The parabola is the locus of a point, whose distance from
a given point is always equal to its distance from a given
fixed line.
Let KX (fig. 26) be the given fixed line, and JS the given
point, from which draw §.X perpendicular to AX, and bisect
itin 4; then A is a point in the curve; and since the dis-
tance SX is known, let it equal 2a, and consequently 4S =a.
Draw Ay parallel to KX, and take 4 for the origin,
and Aw, Ay, for the rectangular axes of the co-ordinates; and
let P be a point in the parabola, on the same side of KY
as S, and AN=x, PN =y, its co-ordinates; then drawing
PM perpendicular to AX, and joining SP,
NW be AY 27 bp
OFM EN eS Vier kds
or y+ (w—a)’=(e+a)’;
“nif =40a,
the equation required.
70. To trace the parabola by means of its equation
(fig. 26).
Solving the equation, we get y = + 2\/aza,
which shews, since for each positive value of x there are two
equal valucs of y with contrary signs, that the axis of v7, Xa,
42
is an Axis of the curve, (Art. 63); and that the origin 4
is a point in the curve, since w=0, gives y=0; and that
no part of the curve is situated to the left of A, for a nega-
tive value of « makes y imaginary; but that as & increases
from zero to infinity, y also increases from zero to +0.
Moreover, as we shall soon see, the tangent at the vertex
is perpendicular to the axis; therefore about the vertex,
and consequently at every point, the curve is concave to-
ward its axis, otherwise it might be intersected in more
than two points by a straight line, which is impossible,
(as will appear, Art. 89). The parabola has only one ver-
tex, namely, the point A where it is met by the axis; and
only one focus and directrix; and consists of two perfectly
similar infinite branches 4x, 4s’, upon the same side of the
axis of y, and situated symmetrically with respect to the
axis of w, to which they turn their concavities.
71. The double ordinate through the focus is called the
latus rectum of the parabola; to find its length, making
e= AS =a, we get
ye sary Oya 2a HS Bor SCs
and consequently BC = 4a.
Hence, if P be any point in the curve, we have
PN*=BCx AN;
or, the square of the ordinate is equal to the rectangle of
the latus rectum and corresponding abscissa, and consequently
varies as the abscissa.
Hence it is easy to determine any number of points in
a parabola, whose latus rectum is known. MHaving taken
SX = Slat. rect. (fig. 26), through any point N in the axis
erect the perpendicular PP’; then with centre § and radius
equal to XN describe a circle cutting the perpendicular in
P, P’; these are evidently points in the curve. To describe
the parabola by a continuous motion, make a right-angled
triangle KMR, having a string, length MR, fastened at R
and the other end at §, slide along the directrix, and
at the same time make a point P slide along RM so as always
43
to confine a portion of string PR against RM; then the
point P will trace out a portion of the parabola, for SP will
always equal PM. If the angle RMK were acute, the locus
of P would be a hyperbola, because SP = PM would be
always greater, in a constant ratio, than the perpendicular
distance of P from the line KX.
72. Let the origin be a point C (fig. 27) in the curve,
and let the axis of the abscissee be a line perpendicular to
the axis of the parabola, passing through C; and let CM = a,
MP=y, be the co-ordinates of any point P, and CB =h,
ABz=k, those of the vertex; then
PN’ = 4a. AN,
or (h — x)’ =4a(k—y), or —2ha + a = — 4ay,
: h a
since A? =4ak; or y=— av —- —,
24a 4a
another form of the equation which is sometimes useful.
73. To express the distance of any point in the parabola
from the focus, in terms of its abscissa.
By Definition, SP= PM (fig. 26),
= XN= XA + AN,
SP=ax2+ a4.
74. In expressing, as above, the distance of any point
In the parabola from an assumed fixed point, it is only when
the latter coincides with the focus that the expression becomes
rational,
For let «’, y’, be the co-ordinates of the assumed point ;
w, y, those of the point in the parabola, and d their distance ;
then
D = (w — 2’)? + (y-y) =v —2aa' + uv? 4 y—2yy + y?;
fir’ Oo ° °
but y =24/ax, therefore d’®, and a fortiori d, cannot be
rational in terms of «#, unless the term 2yy' vanish, which
gives y'=0; then, replacing y’ by its value,
P= a+ 4a0—-2axv + av,
i,
all = ail -
2
ee - SRE
—_———
SS ee
—
cae 7?
eH — &
But the points being in the parabola, we have
yy? =4an", y? = 4aa’;
. yf? —y? = 4a(a” — 2’),
Ea EE
ed i},
ew — ey +y
4a
tana =—,—.
y+y
Now let P’ move up to and coincide with P, then a” = a’,
y" = y’, and the secant becomes the tangent at (w’, y’); there-
SESS
Scents
46
fore, denoting by a the angle PT'N which the tangent makes |
with the axis of 2, we get
4a 2a
°
2y - y
and consequently the equation to the tangent is
m= tana=
, 2a ’
y-y vf aC IBY
or, multiplying by y’ and observing that y? = 4a",
yy =2a(v+ a);
in which a’, y’, are the co-ordinates of the point of contact,
and #, y, co-ordinates of any point in the tangent line; or,
lastly, in terms of its inclination to the axis, the equation |
to the tangent is
aa! au a
yY=Me+ 7 = Me+ sy = 20 -—.
Y m
: 20 ; |
77. If in the formula tana =—-, we make y = 0, we
y
find tana=oo; therefore the tangent at the vertex is per-|
pendicular to the axis. Also if we suppose y’ to increase up)
to infinity, a decreases to zero; therefore the tangent to the’
parabola continually tends to become parallel to the axis.
Hence, the equation of Art. 72 may be put under the
form |
x’
y=axtanp— —,
4a
2
putting 2 7'CB = B (fig. 27); for cot T7CB = a
78. In any curve the distance between the foot of the
ordinate to any point, and the intersection of the tangent at
that point with the axis, is called the subtangent.
In the parabola, the subtangent is double of the ab-
scissa.
47
For 7N x tan PTN= PN (fig. 29),
2a y"
or (Art. 76) T'N x ae y; « TN=
= 2a =2AN.
2a
This result may also be obtained by making y=0 in the
equation to the tangent; this gives w= — 2’, and proves that
the point 7’ where the tangent meets the axis of «, is situated
to the left of A, and at a distance A7'= AN. Hence, adding
AN to AT’, we have the subtangent 7'N = 2 AN.
This property furnishes a simple construction for drawing
a tangent to a parabola at a given point of the curve.
If P be the given point, and 4N, NP, its co-ordinates,
we have only to take in NA produced, AT'= AN, and join
TP, then 7'P is the tangent required.
79. In any curve, a line drawn through the point of
contact perpendicular to the tangent is called a normal; and
the distance between the foot of the ordinate, and the inter-
section of the normal with the axis of w, is called the sub-
normal.
In the parabola, the subnormal is equal to half the latus
rectum.
For if PG be perpendicular to PT’, since in the triangle
TPG (fig. 29), PN is drawn from the right angle perpen-
dicular to the opposite side,
NGxTN=PN’, or NG x 2v=4a4;5
.. NG =2a= half the latus rectum.
80. This result may also be obtained by finding the
equation to the normal at the point (w’, y’) of the parabola.
It will be of the form y — y' = m’ (a — 2’); and since the
normal is perpendicular to the tangent whose equation is
2M
y— y' = y (x = a’), (Art. 76),
48
1 y
wevhave on’ =~ eee (Art. 23) ;
m 24
therefore the equation to the normal is
,
Ula ne
Y SY ae ae (vw — a);
now make the ordinate of the normal y = 0, then w — a’ = 2a,
or AG— AN = NG = 2a.
174)
<
e A] e
Also since y’ = — 2am’, a’ = ric am”, the equation to
a
the normal in terms of its inclination to the axis, is
y + 2am’ = m'(«# —am").
81. Since S7T’= AS 4+ AT =a+2,
and SG= SN+ NG=a-a+2a=a+4+2,
we lave Sie. SJ = SG CAs).
Draw Pe’ through P parallel to the axis,
then 2¢P2’ = PTS = SPT, since SP= ST:
also 2 GPa’ = SPG.
Hence the tangent and normal at any point make equal
angles with the focal distance of that point, and with a line
drawn through it parallel to the axis.
82. ‘These properties furnish a simple method of drawing
a tangent to a parabola through a given point.
First, let the point be in the parabola, as P (fig. 29) ;
join SP, and with centre § and radius SP, describe a circle
cutting the axis in 7’ and G; then if P7' and PG be joined,
they are the tangent and normal at P.
83. Next, let the point be without the parabola, as 7’
(fig. 30); and with centre 7’ and radius 7'S' describe a circle
cutting (as it necessarily must, since J’ is nearer to the
directrix than to the focus) the directrix in two points M
and M’, through which draw two parallels to the axis, meeting
the parabola in P, P’, these are the points of contact; for
the triangles MPT’, T'PS, are equal in all respects, there-
fore £7'PM = TPS, and PT is a tangent at P; similarly,
TP’ is a tangent at P’.
It may be observed that the tangents 7'P, 7'P’, subtend
equal angles at §; for 2 TMP=zTM'P, being comple-
ments of the equal angles 7'MM’', TM'M: therefore,
moe = TSP. Also 2:SPT = PM — compl’. of PMS =
SMM’ = STP’; therefore the triangles S'7'P, ST'P’ are
similar; hence
| (HPS oe NORD. VE
i P. Be and - = ae
SAMMI ny ou tal TASTE Gee
_ 84. The problem of drawing tangents to a parabola
from an external point, may be also solved by means of the
equation to the tangent, as in the case of the circle.
Let h, k, be the co-ordinates of the given external point,
and 2’, y’, those of the unknown point of contact; then
since 2’ and y' must satisfy both the equation to the tangent
and that to the curve, we have, to determine them,
: , iD ,
ky =2a(h+a'), y?=4aa’';
and. if we construct the straight line represented by the
former, considering w’ and y’ as the variables, the points in
‘which it intersects the parabola are the points of contact.
Hence it follows that if a pair of tangents to a parabola
|be drawn from an external point (2, k), the equation to the
‘chord joining the points of contact is
ky =2a(@ +h);
‘for it is the equation to a line which determines, by. its
‘Intersection with the parabola, the points of contact.
To find the angle a between the tangents that intersect
4
Se SS ———
9 or SSS SS == = =
ae
= oI
2 Oe SE Se
50
in a given point we have, if m, m’, be the tangents of the
angles which the touching lines make with the axis,
(1 + mm’)* tan’ a = (m — m’)? = (m +m')* — 4mm’ ,
since m, m’, are the roots of mh —-mk +a=0, (Art. 76).
If a be invariable, then (a +)’ tan’a = k? —4ah, or
(a+h)seca=k’ + (h—- a)’, is the equation to the locus of
the intersection of two tangents to a parabola that include
a constant angle, and represents a hyperbola with the same
focus.
85. The locus of the foot of the perpendicular dropped
from the focus upon the tangent to a parabola, is the line
touching the parabola at its vertex.
Let PT (fig. 29) the tangent at P, meet Ay, the line
touching the parabola at its vertex in Y, and join SY; then
because 7'N is bisected in A, PT’ is bisected in Y since Ay
and PN are parallel (Art. 77); and since SP= ST’, the
triangles SPY, S7T'Y, are equal in all respects ; therefore
SY is perpendicular to PT. Hence the tangent at any point,
and the perpendicular upon it from the focus, intersect in
the line which touches the parabola at the vertex.
Also, from the right-angled triangle SYT', since AY is
drawn from the right-angle perpendicular to the opposite
side, we have
SY? = ST x SA=SP x SA,
-or p’=ar, denoting SP, SY, by r and p respectively.
86. Let the tangent at P (fig. 31) meet the directrix |
in Q, draw PM perpendicular to the directrix, and join |
SQ; then SP=PM, PQ is common to the two triangles
SPQ, QPM, and 2 SPQ = QPM by what has been proved ;
ZQSP = PMQ =a right angle.
51
Hence if a perpendicular through the focus to any focal
distance §'P, intersect the directrix in Q, and QP be joined,
QP is a tangent at P.
Therefore, producing PS to meet the parabola in P’ and
joining QP’, this is a tangent at P’; and since Z PQS = PQM
and £P'QS = P'QM’, we have z PQP’ = ta. Hence the
tangents at the extremities of any focal chord intersect at
. S e e e 7, . e
right angles in the directrix; and the line joinine their
o oO J oO
oint of intersection and the focus. is erpendicular to the
Pp ’ per}
chord.
87. Any circle passing through the points of intersection
of three tangents to a parabola, will also pass through the
focus.
eter a Qs -P (fig. 77) be the three points of contact,
L, M, N, the three points of intersection. Draw SD, SE,
S#', from the focus to the points where the tangents cut
the tangent at the vertex; these are respectively perpen-
dicular to the tangents. Then
LLMN = DTS + FRS=SDE + SFE
=SLE+ SNE=7-LSN:
therefore a circle may be described about the quadrilateral
MLSN.
88. The results in Arts. 85 and 86 may also be ob-
tained as follows. The equation to the tangent at (w’, y’)
being
a 2a
y= mex + —, where'm = —
iG Y
the equation to a perpendicular upon it from the focus is
>
-(w ~ a)
y= — —(« -a);
i m ;
hence the co-ordinates of the point of intersection are wv = 0,
which characterises the tangent at the vertex, and
a y
2
J — he
=
- SY? =a%+y = 0° +a0' = SP. SA.
4—2
WIBRARY
* Fh 8): hraee & is INOIS
INIVERS!! ¥ OF Lin ati
tive ¥ . oye
iD CHAMPAIL :
AT Lino ¥
o2
Next let h, & be the co-ordinates of the point of intersec- |
tion of a pair of tangents, then the chord joining the points |
of contact has for its equation |
ky =2a(e@ +h);
and if it pass through the focus, when 7 = a, y =0;
-. h = —a, the equation to the directrix.
Also the equation to a perpendicular to the chord through
the focus, is
k
y= earn oe and when a= —a, y=k;
therefore the perpendicular passes through the intersection
of the tangents.
Lastly, let a’, y’, v”, y”, be the co-ordinates of the ex-
tremities of the focal chord, then they satisfy its equation;
— 4¢q°
aa 88 or yy ODT res).
consequently. the tangents at the extremities of the focal
chord are perpendicular to one another.
The Parabola referred to Oblique Co-ordinates.
89. To determine the intersection of a straight line with
a parabola.
Let y= ma +c, be the equation to any straight line;
if this line intersect a parabola whose equation is y’ = 4aa,
and if a’, y’, be the co-ordinates of a common point, we have
, /
y=ma+e, y= Aad 3
e e 1 ’ > °
therefore, substituting — (y’ — c) for «’ in the latter, we get}
m |
the roots of which are the ordinates of the points where
the straight line meets the curve, and the abscissz are known
, 1 , e e
from x =—(y'—c). Hence if the roots are real, the straight
m |
line will cut the parabola in two points, and it cannot cut
the parabola in more than two points; if the roots are
imaginary, the line falls entirely without the parabola.
If the roots be equal, the points of section coincide, and
the line is then a tangent; and we have, since the first mem-
ber of the equation is a perfect square,
l6ac 16a? a
—_ =—— , ore=~—,;
m m m
Y= VA + —— ¢
Is the equation to the tangent to a parabola, in terms of
the angle which it makes with the axis, agreeably to Art. 76.
90. To find the locus of the middle points of a system
of parallel chords.
Let QQ’ (fig. 32) be any chord whose equation is y=ma+c,
V its middle point ; draw VM perpendicular to 4, then
VM =QN'+43(QN - Q‘N’) =1(QN’ + QN);
but the values of QN, Q’N’, are the roots of the equation
' obtained by eliminating w between the equations to the para-
bola and chord, as in Art. 89:
A
fe QNGEEORY, = a a
m
hence, denoting the ordinate of V by Y, we have for the
20 :
equation to the locus of V, Y=—, which represents a
m
‘straight line PV parallel to the axis; and since the equation
54
does not involve c, PV bisects all chords for which m is the
same, that is, the system of chords parallel to QQ. :
91. The line PV is called a diameter of the parabola ;
and the semi-chord QV is called an ordinate to the diameter
PY.
¥ e s .
“, Hence all diameters of a parabola are straight lines parallel
to the axis; and conversely, every straight line parallel to
the axis may be considered as a diameter of the parabola ;
2a
for by elving m a suitable value in the equation Y=—,
m
Y may become equal to any quantity we please.
92. Suppose a diameter Pw’ to be drawn at a distance
y’ from the axis; then we have for this diameter
or 7% =
ACG 2a
t
The quantity m is the tangent of the angle at which the
diameter in question meets the chords which it bisects; it
is also (Art. 76) the value of the tangent of the angle which
the line touching the parabola at P, makes with the axis
of x; therefore the chords bisected by any diameter, are
parallel to the tangent at the extremity of that diameter ;
as might have been foreseen; for of the parallel chords which
PY bisects, that which is indefinitely near to P, will ultimately
coincide in direction with the tangent at P. Hence, also,
the diameters bisect the corresponding chords at different
angles varying from a right angle to zero. (Art. 77).
In order that m may be infinite, we must have y =0;
hence the axis of # is the only diameter which bisects its
ordinates at right angles, or is the only axis of the parabola.
93. To-find the equation to the parabola when referred
to the system of oblique axes formed by any diameter, and
the tangent at the extremity of the diameter.
Suppose the new origin to be a point P (fig. 32) in the
curve, and h and & its co-ordinates; then between h and k
5)
e
OV
we have the relation k° =4ah; also let the diameter Pa’
be the axis of #’, and the tangent at its extremity, Py’, the
: 2a .
axis of y’; and let zy’ Pw#’ =a, then tana= a (Art. 76).
¢
Let AN= a2, NQ=y, be the’ co-ordinates of any point Q
reckoned from the vertex as origin; and PV =a’, QV=y’,
its co-ordinates referf#xt to the naw axes; then VR as
RQ=y' sina, and .
AM + MN =h®@# y cosa,
&
Il
y=NR+RQ =k+y'sina.
Now substituting in the equation y? = 4a, we get
(y' sina + k)* = 4a(#' + y’ cosa +h),
or y*sinn?a+ 2yksina+kh =4aw' + 4ay' cosa + 4ah,
2 e ¢ , . °
or y sin’a = 4aa", since kh? =4ah, and ksina = 2a cosa.
e a €
But (Trig. 18) —— = acosec’ a =.a (1 + cot?a)
sin’ @
7)
i
= a (1+ ) =-a+h=SP,
4a~
. y®=49P.a =40' 2’, if SP=a,
or QV* = 4SP. PV.
94. The coefficient 4,8'P, by which one diameter differs
from another, is called the parameter of the diameter to
which the parabola is referred; it is equal to four times
the distance of the focus from the extremity of the diameter.
It is also equal to the double ordinate passing through the
focus. For draw QQ’ (fig. 33) through the focus § parallel
to the tangent P7'’; then PV = ST = SP,
* QV? =4SP x PV =4SP’;
“. QV=2SP, and QQ’ =4SP.
95. ‘The equation to a parabola being of the same
form when referred’ to a diameter and the tangent at its
56
extremity, as when referred to the axis of the parabola, the
properties which are independent of the inclination of the
co-ordinates, will be the same in the two systems. Hence,
taking y?=4a'a# for the equation to the parabola referred
to the oblique axes Pa’, Py’, (fig. 35), the equation to the
tangent at a point Q(a’, y’) will be (Art. 76)
yy =2a («#+2’),
2a’ . . :
where —~ denotes the ratio of the sines of the angles which
Y
the tangent makes with the axes of wv and y (Art. 30); and
when the tangent meets the axis of &, we shall have w = — #’,
or PT’ = PV; i.e. the subtangent equals twice the abscissa of
the point of contact, in all cases.
96. Also, if we wish to draw a tangent through an
external point Q(h, k) (fig. 34), we have, to determine the
points of contact (a’, y’), the equations
y?=40 2, yk =2a' (e +h);
the latter, considering « and y’ as the variables, being the
equation to the chord joining the points of contact; and
if this line be constructed by taking
2ah
PT=-h, PR= —_,
k
and joining 7'R, it will cut the parabola in the two points
of contact.
97. Since the distance PZ’ is independent of k, if |
through Q we draw a line parallel to Py’, and from any |
other point in this line we draw a pair of tangents to the
parabola, the secant passing through the new points of contact
will cut the diameter Pw’ in TJ’, as this point only changes
when # changes. Hence if from the several points of any
straight line, pairs of tangents be drawn to a parabola, the
secants joining the corresponding points of contact will all
intersect in the same point; and.conversely, if through any
point we draw different chords, and draw. two tangents at
57
the extremities of each, the locus of the intersection of the
tangents will be a straight line,
Hence it appears that the same equation ky = 2a(v +h)
represents (1) the tangent at the point (A, k&) of the curve;
(2) the chord of contact of two tangents, drawn from an
external point (h, k); and (3) the locus of the intersection
of pairs of tangents applied at the extremities of all chords
passing through any point (h, k).
98. The tangents at the extremities of any chord will
Intersect in the diameter of which the chord is an ordinate,
For taking the diameter and the tangent at its extre-
mity as axes, the equation to the tangent will be
tL yy =2a' (@ +2’);
using the upper or lower sign, according as we consider the
point Q (a, y’), or the other extremity of the chord Q’, whose
co-ordinates are a’, —y', to be the- point of contact (fig. 35) ;
and in both cases y=0 when # =— 2’; therefore the tan-
gents meet the diameter in the same point 7".
99. Having given the parameter of any diameter of a
parabola, and the inclination of the corresponding ordinates,
to describe the parabola.
Let Pw’ be the given diameter (fig. 33); draw the line
y PT at the given inclination to Pw’, this line will be a
tangent to the parabola at the point P. Make the angle
TPS = y Px’, and PS'= a quarter of the given parameter ;
then § will be the focus. In PV produced backwards take
PM = PS, and draw ML perpendicular to Ma’, this will be
the directrix; and the focus and directrix being known, the
parabola can of course be described.
100. If a parabola be traced upon a plane, we ma
determine its axis by drawing two parallel chords QQ’, ad
(fig. 35), and drawing a line VV" through their middle points,
this will be a diameter. And if we draw any chord QR
perpendicular to it, and through the middle point of QR
58
draw AN parallel to VV’, this will be the axis of the
parabola, and if from P we draw a line making with the
tangent at P an angle equal to y’ Pa’, it will intersect the
axis in § the focus,
101. If through any point within or without a para-
bola, two lines be drawn parallel to two given straight
lines to meet the curve, the rectangles of the segments will
be to one another in an invariable ratio.
Let O be the given point (fig. 36), Qq a line drawn
through it in a known direction, and therefore an ordinate
to a given diameter PV; draw the diameter A’O, and A’v
parallel to Qq; then
QO x Oq = QV?—VO? = 4SP x PV — 4SPx Pv = 4SPx A'0.
Similarly, if Q’q’ be an ordinate to the diameter whose ex-
tremity is P’, passing through QO,
Q'O x Od = 4SP’ x A'0;
wo» QO Oge: 6G O50 Ce sap banoSaes
a ratio independent of the position of O.
If one of the lines be parallel to the axis, then it 1s
the ratio QO x Oq : A’O that is invariable, however Qq
and 4’O move parallel to themselves.
102. Hence if we suppose Qq, Q’q’, to move parallel to
themselves till they become tangents to the parabola at the
points P and P’, and intersect in a point O without the
curve, we have agreeably to Art. 83,
OP? : OP? : SP: SP’.
103. Any parabolic segment ANP cut off by a diameter
and its semi-ordinate, is two-thirds of the parallelogram whose
sides are the abscissa AN and the ordinate NP.
Let NP, N’P’ (fig. 37), be ordinates to the diameter
AN; at P, P’, draw tangents meeting the diameter in T,
59
T’, and one another in R. Join PP’ and draw RK parallel
to AN, meeting PP’ in J, and bisecting it in that point
(Art. 98); and draw KH perpendicular to AN.
Then area of triangle T7RT"=17TT"' x KH,
area of trapezium N’P’PN= NN’ x KH=TT’ x KH,
(Art. 35) since AN= AT, and AN'’=AT’.
Hence the trapezium is double of the triangle. Similarly
we may shew that trapezium N”’P”’P’N’ is double of the
triangle 7”’R'T”, and so on. Hence the sum of the trapeziums
is double of the sum of the triangles; and therefore the
parabolic segment APN, which is the limit of the first sum,
is double of the exterior sezment APT’, which is the limit
of the second sum. Hence the segment ANP is two-thirds
of the triangle Z.NP, or two-thirds of the parallelogram
contained by AN, NP.
SECTION VII.
ON THE ELLIPSE.
Various Forms of the Equation to the Ellipse.
104. To find the equation to the Ellipse.
The ellipse is the locus of a point, whose distance from
a given point is always less than its distance from a given
fixed line, in a constant ratio.
Let S (fig. 38) be the given point, and KK’ the given
fixed line; and from S§ let fall the perpendicular S.Y upon
KK’. Let P bea point in the ellipse; join SP and draw PM
perpendicular to AX, and let the constant ratio of SP to PM
be e : 1, e being less than 1; then P is on the same side of
Kk was (S° Divide SSX in’ Avso\ that "9 4)= eat ee
then 4 is a point in the ellipse; and since the distance
S.X is known, we may assume AS’ = p, therefore TAX oe
e
Through A draw Ay parallel to KX, and take A for the
origin, and Aw, Ay, for the rectangular axes of co-ordinates ;
and let AN =x, PN =y, be the co-ordinates of P, so that
SN =«-p, XN =0+".
Then SP’? =e’. PM*,
or SN? 4+ NP? =e’. XN’,
Il
I
or (v-p)?+y°
» (P ; :
a & + x) = (p + ex)’;
e
.y =2p( +e)rx—-—(1 —-e*) 2’
NV
61
or, if we replace the known quantity a by a,
y = (1 - e’) @aa - 2’),
the required equation.
105. To determine the points where the curve cuts
the axis of 2, make y=0 in this equation; then x =0 or
w= 2a; the value w = 0, gives the point 4 already known;
the other value w = 2a = 44’, determines the point 4’.
Bisect 4A’ in C, then making in the equation to the
ellipse 7 = AC’ =a, we get |
y=-e&)a, or y=+£aV1—e.
If therefore through C we draw BB’ perpendicular to
AA’, and take CB = CB’ =a Vai 6. 7b. B teare points in
the ellipse; and denoting BB’ by 26, we have
Oo
=
ar\/1—e?=b and 1 a ee >
and the equation to the ellipse becomes
b PSS
y= —V 200 — x.
106. In order to transfer the origin to Cy, since
AN = AC + CN, we must change x into a+’; therefore
the equation. to the ellipse referred to its center and axes,
becomes |
o
~
2 b : / V9) 9 ,
y= fea(at a) —(a+ayt = 5-29,
or, in slightly different forms, suppressing the accent,
y
9
~
= i
aes 2.9 nares x
ay” + b°2* = a°b*, or — +
a
This form of the equation shews that the origin is the
center of the ellipse, and that the axes of the co-ordinates
are Axes of the ellipse (Art. 64); but the term, axis, is more
Beuauany appropriated to the portions of those lines,
AA’ =2a, BB' =2b, which fall within the curve; of these
the greater (which passes through the foci, as we shall see)
is called the major axis, and sometimes the transverse axis;
and the other the minor, or conjugate axis. Their ex-
tremities A, 4’, B, B’, are called the vertices of the ellipse,
and their intersection, as has been said, the center.
107. To trace the ellipse by means of its equation.
The equation to the ellipse referred to its center and |
axes 1s
b a See
Y = & —a/@? — v’.
a
Hence as # increases positively from zero to a, the two
values of y are real, and diminish from 6 to zero, and give
the portion of the curve BA’B’ (fig. 38); but when & ex-
ceeds a, the values of y become imaginary, and therefore
no part of the curve lies beyond 4’; also the curve in
every one of its points must have its concavity turned towards
the ‘center, otherwise it might be cut by a straight line in
more than two points, which is impossible, (as will appear
Art. 137). Hence, since the portion of the curve situated to
the left of BB’, is precisely similar and equal to the portion
situated to the right, the shape of the curve is that of the |
oval 4B A’B’, surrounding the center on every side, and every
point in it being at a finite distance from the center.
108. Since the ellipse is symmetrically situated with
respect to the axes 44’, BB’, if we take CH = CS, Cx = CX,
and draw ke pernetienie to AJ’, there is no reason why
the curve may not be described by means of the focus H
and directrix Aa, just as well as by means of S' and KX,
Hence the ellipse has two foci S and H, equidistant
from C.
Also, since (Art. 104) a= = , we have 4S = p = a(1-e);
—e
“. SC = AC - AS =a—a(1—e)=ae, ande = —.
a—a(l—e)=ae, ande eo
|
63
109. The quantity e, which expresses the ratio of the
distance between either focus and the center, to the semi-
axis major, is called the eccentricity.
Since (Art. 105) b=a\/1—e*, the eccentricity e, ex-
Mas —o
pressed by the semi-axes, is equal to
Hence SC = ae =\/a? — 6, and... BS =a;
and if with center B and radius equal to the semi-axis major
we describe a circle, it will intersect the major axis in the foci.
Hence also, each focus divides the major axis into the
segments @ — / a —b*, a+ V/ a? — b*, the product of which
equals the square of the semi-axis minor; that is,
AS. A'S = BC’.
110. Since 4S =e. AX, we have ypc en,
a (1 —e’
and SAY = SA + yee ia, Si
€ CS
which determine the directrix relative to the center, and
focus.
111. The double ordinate passing through the focus
is called the latus rectum.
To find its value, make « = CH =ae in the equation
2
b
y = — (a? — 2°), (Art. 106)
t
2 b*
be
then 7? = 7 (a - ae) =0°(1 -e') = 5 (Art. 105)
64
112. If the distance SC between the focus and center
of an ellipse be supposed to become infinite, the distance
AS between the focus and vertex remaining finite, the
ellipse will be changed into a parabola.
9
For since — = 1 —e’, the equation to the ellipse reckoned
a”
j a
from the vertex, may be written
y =2a(1—-e)vx-(1- e’) a’,
or y =2p(l+e)a—(l—e’)a2’, if AS =a(i —e) =p.
SC MG 1
Art. 108) e = —~ = ——__ = —>-;;
ae AC p+SC ene
TSC
let NC =.00 3). “ie = 1;
“. y =4pa, the equation to a parabola.
Hence, if any result be obtained for the ellipse, we may,
by this modification, adapt it to the parabola ; that is, by
expressing it in terms of 4, and SC, and then making
SC = ©, the origin of co-ordinates being supposed to be at
the vertex or focus.
113. When a =b, the equations to the ellipse become
y =2a%—2", y =a? — x, (Arts. 105, 106),
which represent a circle; hence when its axes are equal, the
ellipse becomes a circle.
Upon the major axis as diameter describe a circle (fig. 39),
and produce the ordinate NP of the ellipse to meet it in Q;
then making CN=a2, NP=y, NQ=y/’, we have
y = /a-ax:
which shews that, corresponding to the same abscissa, the
ordinate of the ellipse is to the ordinate of the circle in the
constant ratio of the smaller to the larger axis; consequently
65
the ellipse may be described by diminishing all the ordinates
of the circle in that ratio.
o
~
a b b°
114. Since y? = = (a — a*)= 5 (a+uv).(a-2),
: b?
gives PN’ =— x A’N x AN (fig. 39),
) a
we see that the square of any ordinate is to the rectangle of
the corresponding segments of the Major axis, as the square
of the semi-axis minor to the square of the semi-axis major ;
and that, consequently, the square of the ordinate varies as
the rectangle of the corresponding segments of the major
axis,
115. To express the distances of any point in the ellipse
from the foci, in terms of its abscissa.
By Definition, (fig. 38),
SP=e.PM=e(CX + CN) =e (E+ 2) (Art. 110)
a + eX.
HP=e.PM'
¢ (Cx - CN) =e (=~ a)
=a-euv;
since e is less than 1, and @ is always less than a, this
expression for HP is always positive.
If w be measured from §, then
SP=e.PM=e(SX+ SN) =a(1—-e’) + ex.
116. In expressing, as above, the distance of any point
0 the ellipse from an assumed fixed point, it is only when
the latter coincides with one of the foci, that the expression
decomes rational in terms of the abscissa of the point.
For let wv’, y, be the co-ordinates of the assumed point,
’, y, those of any point in the ellipse, and d their distance,
66
then @ = (a —a') + (y-y')’
, / 19
=a —2eu +n"? +y°—2yy +y-
b 2 y >) > ° .
But y=- \/ a — x, therefore d’, and a fortiori d, cannot
a
° . ,
be expressed rationally in terms of 2, unless the term 2yy
disappear, which gives y’=0; then replacing y” by its value,
we get .
b° aan ts
d’ = (1 - =) ao” —2na' + a” +d,
ae
which must be a perfect square ;
b°
14 (1 - =| (av? + 6") = 42”,
or v = + \/a? — B?.
These values require that a should be greater than 6,
i.e. that the abscissee should be measured along the axis
major; and with y’ =0, they determine, as we perceive, the
two foci S and H. These then are the only points whose
distances from every point of the curve can be expressed
rationally in terms of the abscissa, or rather of the co-ordi-
nates of the point. For, relative to any origin and axes
whatever, we should have SP=a+e(ma+ny+h).
This is sometimes given as the definition of the focus.
If the co-ordinate axes were turned about the focus through
an angle @, the formula \/a* + y? =a(1—e’) + e# would be
transformed into
f/x +y2=a(1— e’) + e(#’ cosP—y' sin®@)=c + ma +ny.
Hence we see that a®°+y?=(c+ma+ ny)’ represents
an ellipse with its major axis inclined to the axis of w at
nv e e . e
an angle whose tangent = ——, and of which the origin is
m
one of the foci, provided m* + n? <1.
67
117. Hence, by addition we get
SP + HP =2a,
or, the sum of the focal distances of any point in the ellipse
is constant, and equal to the major axis. Also, for a point
not in the curve, the sum of the focal distances is greater
or less than 2a, according as the point is situated outside
or inside the ellipse (Eucl, 1. 2
This property affords a simple method of determining
any number of points of an ellipse of which we know the
foci and axis major. In 4d’ take any point # (fig. 40), and
with centre § and radius A’F describe a circle; next with
centre H and radius AF describe another circle, cutting the
former in P, P’, these are manifestly points in the ellipse.
When the ellipse is to be very large, we may describe
it by fastening in the foci the ends of a cord of the same
length as the axis major; then if we make a pole slide along
the cord so as to keep it stretched, the ellipse will be traced
out by the extremity of the pole.
This property also furnishes the following method of in-
vestigating the equation to the ellipse.
118. To find the locus of a point, the sum of whose
distances from two given points is constant.
Through the two fixed points S, H (fig. 40), draw the
indefinite line Sw; bisect SH in C, and draw yC perpen-
dicular to it, and take Ca, Cy, for the axes of the co-ordi-
nates, as the locus of P will evidently be symmetrical with
respect to these lines. Let SC = CH = e, CN=a, NP =y,
the co-ordinates of any point P, and SP + HP = 2a;
then SP? = (7 +c)? +’,
HP’ = (x-c)?+y’;
“ SP? — HP*, or 2a(SP — HP) = 4c2;
2x
fs ad Pg ee
a
SS
=
ESS
————
a
FSS
SSS SSS Ss SSS SSS = = = =
5 sy Ss = ee ———— — = <== ~
= : = - Se San = -
Alec ae 8 Freee
San ee ae eR TP
68
but :-SP+ HP =24a; |
C@#
aze SP=a+—;3
a
ca\? .
(« + =) =(«# +c) +y’;
oF 0
rod ~
4+ Qca t+ Ha + 2er+er+y’,
a
pid aa pe at— Cc:
up =e = 8 — 0) Or ye = =, (a? — x*).
Fe
a
Now SP + HP is greater than SH, or @ is greater than
c, therefore a®—c’ is a positive quantity ; hence, comparing
the above with the equation to an ellipse
b?
; Sie ale rs a”),
we see that the equations are identical, and consequently so
are the curves which they represent, if b? = a? —c*®; therefore
the required locus is an ellipse whose major axis is 2a, and
minor axis 24/@ — Cc.
119. To find the polar equation to the ellipse, one of
the foci being the pole.
Let the polar co-ordinates of any point P be SP=7,
ZuSP =60 (fig. 38); then
SP =e.PM =e(XS+SN),
é
or ¥ =. jee + 1 COS a| (Art. 110) ;
r(1 —ecos0) =a(1 —e*),
We have measured the angle @ from that part of the
axis major which passes through the vertex furthest from
69
the pole; sometimes it is measured from the nearer vertex
A, in which case if ASP = 6’, putting +-—6' for @ we get
a (1 — e’)
~ 1+4ecos’-
Of course if the other focus H be taken for the pole,
the formule will be exactly the same.
120. To find the polar equation to the ellipse, the
centre being the pole.
Let CP=r, 2 ACP =6 (fig. 40), be the polar, and a, y,
the rectangular co-ordinates of any point P; then # = rcos @,
y=rsin@; therefore substituting these values in the equation
which, since — = 1-—-e*, may be written
a
y’ + (1 —e’) a = b*, we get
r” (sin® @ + cos? @ — e® cos 0) = B’,
or r°(1 —e’ cos*@) = 8;
b
~ 4/1 — e cos? 0
In this formula it is indifferent from which vertex the
angle @ is measured; it shews that of all lines drawn
from the centre to the curve, the semi-axis major is the
greatest, corresponding to @=0; and the semi-axis minor
the least, corresponding to 0 = 47x.
ey
To get the polar equation from the centre in terms of
the semi-axes, we must substitute rcos@ for w, and rsin@
for y, in the equation a’y’ + b’a* = a’b? and the result is
7 (a* sin’ 9 + 6° cos’ 9) = a°0’.
Tangent and Normal to the Ellipse.
121. To find the equation to the tangent of an ellipse
at a given point.
As in former cases (Arts. 55 and 76), we shall regard
the tangent as a secant which passes at first through two
points of the curve, and then turns about the given point ~
till the other point moves up to and coincides with it; so —
that if m be the tangent of the angle which it ultimately
makes with the axis of #, and a’, y’, the co-ordinates of
the given point, its equation will be
y-y =m(e—2'), (Art. 20)
where m is to be found in terms of # and y.
Let wv”, y”, be the co-ordinates of a point in the curve
near the proposed point, and a the angle which the line
joining them makes with the axis of 7;
, ”
G) eee,
then tana = ;—,,.
C—@&
But the two points being in the ellipse, their co-ordi-
nates must satisfy its equation ;
2 Q 2 2 2 27.2
ey? + Vat = al, ay + 0a” =a°b’.
Subtracting the latter from the former, we get
a? (y? — y'") + b? (x” — a”) =0,
2 , uy)
e h . / Y ee Y b av oe &
WHICh/Olvesptal gas -— =
v —@ ayr+y
Now let the second point move up to and coincide with
the first, then #’ = 2’, y’ =y’, and the secant becomes a
tangent at (v’, y’); therefore, denoting by a the angle which
the tangent makes with the axis of x, we get
ba!
0?
ay!
m=tana=-
7A
and consequently the equation to the tangent is
4
or, under another form, recollecting that a?y? + b’a’? = a®b’,
ayy + bee’ = ab’;
in which 2’, y’, are the co-ordinates of the point of contact,
and #, y, co-ordinates of any point in the tangent line.
122. The formula m= tana = - since it does
not change when «#’ and y’ are replaced by — a’ and —y/,
shews that if PC be produced to meet the ellipse in P’
(fig. 39), the tangents at P and P’ are parallel; as we
might have foreseen from the symmetrical position of the
ellipse, relative to the axes.
Also it proves that at the points B, B’, for which w’ = 0,
y = +b, tana = 0, or the tangents are parallel to 44’; and
at A, A’, for which w#’ = £a and y' =0, the tangents are
perpendicular to 4A’; and that for intermediate points, going
from A to B, the angle P7'x, which is always obtuse, con-
tinually increases till P7’ becomes parallel to 4A’ at B.
123. It is sometimes convenient to have the equation
to the tangent expressed in terms of the angle which it
makes with the major axis. The equation last written down
(Art. 121) gives
b?
Y=MV +-;53
y
bB? a’ am 2 b? xv”?
but from m= ——-—, “we get 2 AEE =i
ay b ay
abt —aty? Me Yeh a
MADRS ToT TT => Taisen tte Cease 5
a’y y” y
“YY =Mce/P + ma’,
the lower sign referring to the point P’.
124. To find where the tangent meets the axis major,
put the ordinate of the tangent y = 0 in the equation to the
tangent; then Uva’ =a'b’,
a CA
2 pan ¢ CT eee
v CN
As this result is independent of 5, it will be the same for all
ellipses constructed upon AA’ as an axis; consequently, if
NP meet the circle whose diameter is 44’ in Q, the tangent
to the circle at Q will pass through 7’.
Similarly, putting # = 0, to find where the tangent meets
the axis minor, we get from the equation to the tangent
9
b :
y=—, or Ct.PN= BC’.
Y
Subtracting CN or wv from the value just found for CT,
we get the subtangent
Ga
NT =— — a =)- = es
v
125. To find the equation to the normal at any point
of an ellipse.
It will be of the form y —y' = m'(v— a’), 2", y', being
the co-ordinates of the proposed point; and since the normal
is perpendicular to the tangent,
therefore the equation to the normal is
OE ath
a”
b? oe ° (w ard x").
v
yY-YyYr
This equation, in the same way as for the tangent, may
be expressed in terms of m’; for we get
y- me + m' (
2
ERP ey rE a
“ JS a +m’*6? = —-;
4?
v
“ (y —m'x) fa + mh +m! (a? - 6°) = 0.
126. The normal at any point bisects the angle between
the focal distances of that point.
First, to determine the point G, where the normal meets
the axis major, make the ordinate of the normal y = 0, in the
equation to the normal, .
b°
5 ham (1 -=) , or CG =e.CN (fig. 41);
*
2
b?
and w — wv = — 2", or the subnormal GN = —. CN.
a” a®
Hence SG=SC + CG =ae+ ew =e(a +e2’) =e.S§P,
and HG = SC —CG=ae-e' x =e(a—ex’)=e. HP, (Art. 115)
NE Se
SHG AP.
and consequently the normal PG bisects the angle SPH
(Euc. vi. 3). Also, drawing the tangent YZ at P, since
4GPY=GPZ, each of them being a right angle, and
ZGPS = GPH; therefore 2 SPY = HPZ, or the focal dis-
tances make equal angles with the tangent on the same
side of it; in other words the tangent bisects the exterior
angle between the focal distances.
127. Drawing GL perpendicular to SP we get from
similar triangles
SG
SL = op SN =e (ae4 X) 5
PL=SP-SL=a+ex—e(ae+2)=a(1-€&),
which shews that if from G the foot of the normal at P'
we draw GL perpendicular to either focal distance, then |
PL =1 the latus rectum. Also
PL PL 1+ecosA SP
tan SPY = cot SPG = See rere aid a
GL e.SP.sinA SP esin A'S P
which determines the angle at which the focal distance cuts
the ellipse.
If we call ST =7, AST = 0, ASP =a, and SPT = q,
(fig. 30) we have
SP sn STP sn(od+a-8)
r sinSPT sin co)
esina SP
=cos(a—0)+sin(a—6). ae ie }cos(a—0) + ecos0} aan
ACL aD Li |
~ cos (a — 0) + ecos 0
= cos (a—@) + sin(a—@)cotd |
the polar equation to the tangent of the ellipse, which is |
sometimes of use.
128. These properties furnish a simple method of draw-
ing a tangent to an ellipse through a given point.
First, let the point be in the ellipse as P (fig. 42).
Join SP, HP, and produce SP to K, making SK =2 AC;
join HK and draw PZ perpendicular to it, then PZ is a
tangent at P.
For in the triangles PHZ, PKZ, PK =2AC —SP =PH,
PZ is common, and the angles at Z are right angles,
LHPZ = KPZ = SPY,
and consequently PZ is a tangent at P.
Next, let the point be without the ellipse.
With S the focus furthest from 7’ the given external
75
point, as centre, (fig. 43), and radius = 2.4C, describe a circle
Kk’; and with centre 7’, and radius equal to T'H the
distance of 7’ from the other focus, describe a circle cutting
the former in AK, K’; join SK meeting the ellipse in P,
and join 7'P; then 7’P is a tangent at P; for in the
Seeies 2 PK PH, PK = PH, ThK=T7H8, and JP is
common, therefore 7'P bisects the exterior angle HPK, and
is a tangent at P; similarly, if SK’ be joined, it will meet
the ellipse in P’ a second point of contact.
As long as T' is exterior to the ellipse, the circles must
intersect. For if S7>2AC, T and # fall on opposite sides
of the circumference KK’; but if 7’ be less than 2.4C, join
ST’ and produce it to meet KK’ in 7"; then S7+TH > ST’
(Art. 117), therefore 7H >T7'T"; and therefore in both cases
the circles intersect. If AK’ be joined, it is evident that
LTKP=TK’'P’; therefore THP = T'HP’; or the tangents
drawn from an external point subtend equal angles at either
focus.
129. ‘This problem may be also solved by means of the
equation to the tangent. Let h, & be the co-ordinates of the
given external point, and a’, y’, those of the unknown point
of contact ; then since a and y’ must satisfy both the equa-
tion to the tangent and that to the ellipse, we have, to de-
termine them,
Pha’ +aky =a'b’, ba? + a®y? = ab.
It is evident that #2 and y’ will each have two values,
therefore there will be two points of contact; and if we
construct the line represented by the former of these equations
regarding # and y’ as the variables, it will intersect the
ellipse in the points of contact.
Hence the chord joining the points of contact of two
tangents drawn from a point (h, k), has for its equation
Pha +aky=a'b’,
9
e e ‘4 a
and it meets the axes of the ellipse in points for which v = i?
:
ae
4
SE ——————
——
——
a a er =
= == — ee ee 5
:
b* ,
Vins which values shew that if two tangents be drawn
from any point in a line parallel to either axis, the chord |
joining the points of contact will pass through a fixed point |
in the other axis, and conversely.
To find the angle a between the tangents that intersect
in a given point we have, if m, m’, be the tangents of the |
angles which the touching lines make with the axis,
(1 + mm)’ tan’ a = (m — m')? = (m +m’)? — 4mm’,
haste Wen Qhk \? ke —
or (1 + fo] tan°a = (=) — A —_——_—
h* -—a
2 We iw
since m, m’, are the roots of (h® — a”)m? -2hkm +k? -B=0
(Art. 123). If @ be invariable, then
(h? +k? — a® — 8°) tan? a = 4(h° 8? + kh’ a? — 70")
|
is the equation to the locus of the intersection of two tangents |
to an ellipse that include a constant angle.
130. The locus of the extremities of the perpendiculars
dropped from the foci upon the tangent to an ellipse, is
the circumference of the circle whose diameter is the axis
major. Produce any focal distance S'P (fig. 42) to K, so
that SK =2AC, join HK, and draw PZ perpendicular to
it; then PZ bisects HK, and is a tangent at P (Art. 128).
Join CZ, then since S'H is bisected in C and AK in Z,
CZ is parallel to SK and equal to } SK = AC. Also, drawing
SY, CQ, parallel to HZ, CQ bisects YZ perpendicularly,
and therefore CY =CZ= AC. Hence the intersections of
every tangent with the perpendiculars upon it from the foci,
are at a constant distance AC from the centre of the ellipse;
or are situated in the circumference of the circle whose
diameter is the axis major.
131. Since C is the centre of the circle which is the
locus of Y and Z, and SYZ is a right angle, and therefore
| 77
jn a semicircle, if YS and ZC be produced to meet in H’,
this will be a point in the circle; and from the equal triangles
hOZH, CHS, SH = HZ;
» SY x HZ = SY x SH’ = AS x A'S = BC’ (Art. 109).
Aico laacs Oo a LZ SY. ASBMB fy
50, SIUCes =. = fr ——_ = —— yy:
: Ree WP? a 7a TP
multiplying this equation by the preceding, we get
S¥8= BO x 2;
HP
or, if SY be denoted by p and SP by 1, and consequently
HIP by 2a—7, we have, between the radius vector of any
point and the perpendicular on the tangent at that point
from the focus, the relation
7
2
29a-7r
p* = b
Draw CE parallel to PY, then PC is a parallelogram ;
therefore PE = CZ= AC; which shews that the portion
of any focal distance cut off by the diameter parallel to the
tangent at its extremity, is invariable.
132. The preceding results may be also readily obtained
by means of the equation to the tangent (Art. 123). For
the equations to P7’ and HZ are, respectively,
Y= Me+ J/ ma? +b’,
if ae ey ge
y=-—(a-Va-b'),
between which if we eliminate m we shall obtain the equation
to the locus of their intersection.
These equations may be written
y —ma =r/ mia? + 0’, a+my=/a— 0;
and adding their squares, the result is
(a? + y’) (m? +1) =a? (m? 41), or a’ +y? =a’,
the equation to the locus of Z.
Again, since HZ is the perpendicular dropped from a
point whose co-ordinates are a’ = / a? — b, y’ =0, upon a
line whose equation is y= ma +4/ ma? + 6°, we have (Art.
28),
¥ —m\/ a — nary SOT E +o
V1+m :
similarly, SY = mV/ a" Vi mia any :
WA 1 +m?
mM? a” e b? head m? (a? Fs b*)
te WELZ, XS Vim aa Fea
m* +. 1
HZ
133. Draw HI parallel to YZ (fig. 42) and let
ZLSPY =a;
SI rsina —-(2a—r)sina
IH rcosa+(2a—7) cosa’
then tan S7'P = tan SH] =
2(7 — a) SP \
OY Cano fies eat Pits oon. y.
; ana (Fe 1) tan SPY ;
~
a result which is sometimes of use.
1S4 Phe tangents at the extremities of any focal chord
intersect in the directrix; and the line joining their inter-
section with the focus is perpendicular to the chord.
Let h, k, be the co-ordinates of the intersection of the
tangents; then the equation to the chord joining the points
of contact is (Art. 129)
he + a’ky = ab’,
and since it passes through the focus, when x# = ae, we must
a
have y=0; therefore h = 5? the equation to the directrix.
79
Also the equation to a perpendicular to this chord
through the focus is
Kia: a ka
y=-—(e-ae); and when w=-, y=—=h,
e
he
therefore it passes through the intersection of the tangents.
Also, if HP =r, HP’ =7, HZ=c, AHP = 0, (fig. 41),
r+r
then tan PZP! = ¢.— 3
eC-—rr
iPr Q a’ (i —e’)’ a (1 — e
but —- + —= = ——___,, rr =~ Cr : 3 Be Mart,
ren rawr (le e") 1 — e’ cos’ 0 esin @
2e sin 8
therefore, substituting, tan PZP' = meEeTS
~— e”
135. If a perpendicular be dropped from the centre
upon the tangent at any point of an ellipse, making an angle
@ with the axis major, its length = a V/1 — e sin? Q-
Let CQ, HZ (fig. 44), be perpendiculars dropped upon
PT the tangent at any point P, from the centre and focus;
join CZ and let 2 TCQ=¢;
then CZ =a, and QZ = CHsing = aesin ~;
C@ =a - ae sin? d, or CQ=a W/o sin’ @.
Hence the polar equation to the locus of the foot of the
perpendicular dropped from the centre upon the tangent to
an ellipse, is 7 =a /1 — e sin? op.
Also if P’7” be another tangent to the ellipse at right
angles to the former, and CQ’ perpendicular to it; then
QCT' =17-; and therefore CQ” =a’ — a’e* cos” dp.
Hence CW? = CQ’? + CQ? =a +a (1-e) =a? +b,
and therefore the locus of the intersection of two tangents to
an ellipse at right angles to one another, is a circle whose
centre is C and radius equal to r/ a? + b%.
These results may be also obtained in the following
manner. ‘The equation to PT’ being
Y—- Me = / ma? + 6° (1),
; : 1 bake ies
the equation to CQ is y= ——w; and eliminating m between |
m
these equations, i.e. substituting in the former —— for m,_
we get for the locus of Q the equation y? + a? = J ax? + b*y”. |
Again the equation to P'W is my + # =4/a? + m°b?; and
adding the square of this to the square of (1) and dividing —
by m?+1, we get a +y’=a*+ 0’, for the equation to the |
locus of W.
‘ 136. It is sometimes convenient to have the length of the -
normal to an ellipse, and also the co-ordinates of the point
where it meets the curve expressed in terms of the inclination
of the normal to the major axis.
From similar triangles (fig. 44), we get
PG Ct
PN CQ’
Ct. PN be?
Pe oi ee (Artin 24candelgs):
CQ an/1 -e* sin’
ahs
Also PN = 6° sin
ar/1—é sin’d
: V1 — e sin’ ar/ 1 — e’ sin’ d
137. All chords of an ellipse which subtend a right
angle at a given point of the curve, intersect one another
in the normal at that point.
Take the given point for origin, and the normal and
tangent at that point for axes of vw and y; then the equation
81
to the curve (which includes every species of conic section)
will be
ay’ + bay +ca° + dy+eu+f=0;
and since the axis of y is a tangent at the origin, if # = 0,
the values of y become each = 0,
. d=0, f=0, and the equation is reduced to
ay’ + bey + ca? + ex =0.
Let y= m(# —h) be the equation to any chord meeting
the normal at a distance h from the origin; then for the
points of intersection with the curve,
am* (x —h)’ + bma (« —h) + ca’ + ev = 0,
of which equation, if ~,, X25 be the roots,
am’ h?
: am +bm-+e
Also eliminating a,
2
ay? + by (% +n) +¢ (2 +4) +re(2 42) =0;
m m m
and if y,, y2, be the values of y,
cm h? + em*h
Sia F iis a ee ?
am +bm +e’
if now the chord subtend a right angle at origin, the lines
joining its extremities with that point, whose equations are
Y1 Yo
y= —x2, y=~~2, will be perpendicular to one another;
0s
*. Pe —, OF 1X2 + YyYo = 05
vy Ye
2 h hi } ° h é
° e = 0, which gives 2 = —
aioe Ch +e : g raps
a value constant for all values of m; hence all such chords
pass through a fixed point in the normal.
82
The Ellipse referred to its Conjugate Diameters.
138. To determine the intersection of a straight line |
with an ellipse. |
Let y= ma+c be the equation to any straight line; if
this line intersect an ellipse whose equation is |
Cy og a1 0.
and if a’, y’, be the co-ordinates of a common point, we have |
y=me +e, @y? +a" = ad’;
3 S ® 1 , , . )
therefore, substituting — (y’ —c) for a in the latter, we get
m
(a?m? + b*)y? — 2b’cy’ + (ce? — ma’) b’ = 0,
the roots of which are the ordinates of the points where the
straight line meets the curve, and the abscissee are known
, i , . .
from « =—(y'—c). Hence if the roots be real, the straight
m
line will cut the ellipse in two points, and it cannot cut the
ellipse in more than two points; if the roots are imaginary,
the line falls entirely without the ellipse.
Ozns. If the roots be equal, the points of section coincide,
and the line is then a tangent; and we have
(am? + 0°) (2 — m?a®) = b’c*, or =O? + ma’;
Se ihe Jb? + ma
is the equation to the tangent to an ellipse in terms of the
angle which it makes with the major axis, agreeably to
ATU Leo.
139. To find the locus of the middle points of a system
of parallel chords.
Tet the chords be parallel to a line CD (fig. 45) through
the centre, whose equation is y = ma; then the equation to
83
any one of them QQ’ is y = mx +c; and the values of QM,
QM’, are the roots of the equation,
obtained by eliminating x between the equations to the chord
and the ellipse, as in Art. 138.
If therefore V be the middle point of QQ’, and CN = X,
VN = Y, its co-ordinates, so that VP = mX + Cs
bc
Y = 4(QM + Q’M’) hye 72?
2 m a” + 0
ay aL. m a” C
iy =—(¥-c)= Saat ea. eer takes
mM mm” a” + lie
therefore, dividing one result by the other in order to eli-
minate the quantity ¢ which particularizes the chord, we get
b?
Y= -—-—..X,
ma
a relation between the co-ordinates of the middle point of
any chord, and therefore the equation to its locus, which is
consequently a straight line PP’ passing through the origin.
The straight line which passes through the middle points
of a system of parallel chords is called a Diameter.
Hence all diameters of an ellipse pass through its centre ;
and, conversely, every line through the centre may be con
sidered as a diameter.
Hence denoting the equations to any chord QQ’, and
to the diameter PP’ which bisects it, by y=mu+ce,y=m'e,
respectively,
6? 6?
we have m’ = —-__, or mm’ = ——,
m a”
a simple relation, by means of which the equation of a dia~
Meter may always be deduced from that of any chord which
it bisects, or vice versa.
§2=9
84
140. If a diameter PP’ bisect the chords parallel to |
a given diameter DD’, then likewise the diameter DD’ will
bisect the chords parallel to PP’.
Let y= ma be the equation to the given diameter
b?
DD’ (fig. 45); then (Art. 139) y = — ag 9 OF
? @
b?
y=me, if m = —-—,
ma
is the equation to the diameter bisecting the chords parallel |
to DD’, which diameter by supposition is PP’. Now let
y = nex be the equation to the diameter bisecting the chords |
2
4
parallel to PP’; then » = — pros Olt aie therefore DD’
a |
is the diameter bisecting the chords parallel to PP’.
Hence two diameters, whose equations y = ma, y=m'a,
2
are so related that mm’ = ——, have the property that each
me
bisects the chords parallel to the other.
Two diameters, which thus mutually bisect the chords
parallel to one another, are called Conjugate Diameters, or
rather those portions of them PP’, DD’, which fall within
the ellipse are usually called a pair of conjugate diameters.
141. If PT be the tangent at P, and a’, y’, the co-
ordinates of P, then the equation to PT' is (Art. 121)
wh ae yah
, ,
-y =-—~ (@-vwv);
Did emicna aeD
e e y e
but the equation to CP is y =—, x, and therefore the equation
v&
2 /
° e ; fe e} b e |
to CD, the diameter conjugate to CP, is y= — —— x, which
er. ay
represents a line parallel to PT".
85
Hence the tangent at the extremity of any diameter,
is parallel to the corresponding conjugate diameter; and if
tangents be applied at the extremities of a pair of conjugate
diameters, they will form a parallelogram circumscribing the
ellipse (Art. 122).
This result might have been foreseen; for DD’ being pa-
rallel to all chords bisected by PP’ is parallel to that which is
situated indefinitely near to P and which ultimately coincides
in direction with the tangent at P when the two points in
which it meets the curve become coincident in P.
142. Having given the co-ordinates of the extremity
of any diameter, to find those of the extremity of the diameter
conjugate to it.
Let «’, y’, be the co-ordinates of the point P (fig. 46),
a ,
y wv is the equation to CP,
&
then y =
nan ;
and .. y = — ——, w the equation to CD.
ay
To determine the co-ordinates of D, we must combine the
equation to CD with the equation to the ellipse a?y’ +b’? 2’ =a?6’,
on
a
which gives, eliminating y by the substitution —
a,
sine F
Cie ae
J
op ge ae RE dg
a YY
or x (Bx? + ay”) =a'ty?, or xa?b? = a! ys
hey ay’
a = 207 Oey — CM = re:
b? a’ ba’
and y = ite —v%=DM=—-;
a
the other pair of values of # and y having reference to D’.
Hence, if we suppose the ordinates NP, MD, produced to
86
ca bay,
meet the circle on the major axis in p, d, Np = hae CM,
and Md = CN, and consequently the angle pCd is a right
angle.
143. The sum of the squares of any two semi-conjugate
diameters is equal to the sum of the squares of the semi-axes.
CP? = CN’? + NP* = x" + y”, (fig. 46)
a y? b? uv?
CD? = CM’ + MD* = pat eee as (Art. 142)
; a
Hog y”
CP? + CD* = (a + 0°) & + 7 =a’ + b’,
a 2
wy
since — + —-=1.
a* b
144, All parallelograms whose sides touch an ellipse at |
the extremities of a pair of conjugate diameters, are equal |
to one another.
Draw CQ (fig. 46) perpendicular to the tangent at P.
Then the area of the whole parallelogram
= four times the parallelogram DP
=4CD.CQ =4CD.CT sn CTQ |
=4C7T.CDsin DCM =4CT.DM
a ba’
=4.—.— =4ab (Arts. 124 & 142).
a
oP
v
145. If we denote CP, CD, by a’, b', and 2 DCP by
ry, and draw PF perpendicular to DC produced, we have |
GO Pi =a sins), |
“0 @ Desin ry = CD ioe a.
This equation, together with a” + 6” = a’ + 6’, determines |
the magnitudes 2a’, 26’, of two conjugate diameters that in- |
87
clude a given angle y; and their position is known from
the equation
CQ=a'siny =a Aone | (Art. 135),
where @ = QCN = —- DCM,
hicl DCM =< Vis de
sh gives s = Spree ie
which gives sind = cos ri ain
If a = 0’, then
Oe ora 2
af or cot@ = tan DCM =
a+b ie
and sin o =
aati:
as might have been foreseen ; for the equal conjugate diameters
being symmetrically situated with respect to the major axis,
b? b?
mm’ = — — gives tan? DCM = —.
2 me
Hence the equal conjugate diameters of an ellipse are parallel
to the chords joining its vertices; and the angle between them
. a . ° . e
is 2tan~’—. In this case, sinry receives its least value; for
it is least when 2a’b’ =a? +b°—(a'—0’)? is greatest; i.e.,
when a@ = 0’. Hence the obtuse angle between two .con-
» . ° is
_jugate diameters varies from . the angle between the axes, to
~
a
2 “lie the angle between the equal conjugate diameters.
146. If we denote CQ or PF by p, we have
a’ b? a’ b?
2 -= — Art. 143
P CDF Oe bt are ( 3)
a relation between the central distance a’ of a point, and the
38
perpendicular p upon the tangent at that point, let fall from |
the centre.
147. The rectangle contained by the focal distances of |
any point is equal to the square of the corresponding semi- |
conjugate diameter.
CD* = a’ + b° — CP* (fig. 40) (Art. 143)
19)
but CP? = 2? 4+yY=074+0 -—- —2¢ = B+ ea’,
e
. CD? =a@ - ex’ = (a+ en). (a—ex)
SP.HP (Art. 115).
148. ‘To find the equation to the ellipse referred to the
system of oblique axes formed by any pair of conjugate |
diameters.
Let w and y be the rectangular co-ordinates of any point
Q in the ellipse referred to its centre and axes; then the |
relation between them is .
ay’ + bx = ab’.
Let the conjugate diameters CP, CD (fig. 47), be the new _
axes of w and y’, inclined to the axis of x at angles PCA = a,
DCA = B, and CV = a’, QV = y’, the new co-ordinates of Q; |
then since the origin remains unaltered, the formule for |
passing from the rectangular to the oblique axes are (Art. 42), |
, , . °
w=# cosat+ycosP, y= sina+y’ sin PB.
Hence, substituting in the above equation and reducing,
(a* sin°a + b° cos’ a) w? + (a® sin® 3 + 8 cos? ) y”
+ 2(a* sina sin 8 + 6° cosa cos B) ay’ = a®b’.
b?
But tan a tan = — — (Art. 140);
a
. @ sina sin 3 + 6° cosa cos 3 = 0,
89
and the term involving «’y’ disappears.
Also if CP =a’, CD = 0, we have (Art. 120),
a” (a? sin’ a + 6° cos’ a) = ab’,
b” (a’ sin? 3 + b* cos? 3) = a°b°;
therefore, substituting and dividing by a*b®, we get for the
| required equation,
fe 72
te!
m7
or, in a geometrical form, supposing PC produced to meet
rhea
the variables, is — +
| a
the ellipse in G, so that PV = a’ —- a’, and VG =a' +4’,
CD
Vy? =
a= Cp
sd AMM Oe
149. This equation, which, suppressing the accents o
9 9 ‘
y”
ia 1, being of precisely the same form
_as that relative to the axes, it follows that all properties which
do not depend upon the inclination of the co-ordinates, will
be common to the Axes of the ellipse and to its conjugate
| diameters.
Hence, w’, y’, being the co-ordinates of any point Q re-
ferred to the conjugate diameters CP, CD (fig. 47), the
equation to the tangent at that point will be (Art. 121)
a°’yy +b? xa =a?b";
| and if the tangent meet the co-ordinate axes in 7’, ¢, we shall
‘have, as before, (Art. 124),
a
i)
nw
C
150.
Also, if we wish to draw a tangent through an
external point Q (fig. 48), whose co-ordinates are h and k,
we shail have, to find the points of contact, the equations
fo
a*y?
19 , loz?
a hey’ + b*ha’ = ab? ;
9 / , Ve
+ b° a? = ab”,
90
the latter (considering w and y’ as the variables) being the,
equation to the chord joining the two points of contact.
And those points, as in preceding similar cases, may be
determined, by constructing this line; that is, by taking |
12 12
CT = 2 CR=—, and joining RT, which will cut the
D
ellipse in the two required points.
Since the distance CJ’ is independent of &, if through |
Q we draw a line parallel to CD, and from any other point |
in this line we draw a pair of tangents to the ellipse, the
secant passing through the new points of contact, will cut.
the diameter CP in 7, as this point only alters when h |
alters. Hence if from the several points of any straight |
line, pairs of tangents be drawn to an ellipse, the straight |
lines, which join the corresponding points of contact will all |
pass through the same point; and conversely if through any |
point (2, &) we draw different chords, and apply two tangents
at the extremities of each, the locus of the intersection of |
the tangents will be a straight line having for equation
a’ ky +b’hxe =a’b". If the line be the directrix, then as |
we have seen (Art. 134), the fixed point will be the focus.
151. The tangents at the extremities of any chord will |
intersect in the diameter of which the chord is an ordinate.
For, taking that diameter and its conjugate as the axes |
of w and y, the equation to the tangent will be
fe ‘
b? va! £a?yy’ = a"b”?,
where the upper or lower sign is to be used, according ,
as we consider the point Q(#’, y’) (fig. 47), or the other |
extremity of the chord Q’ whose co-ordinates are w#’, — 4’,
to be the point of contact; afd in both cases when y = 0,
’9
e
«© =—, therefore the tangents meet the axis of v in the same |
Vv
c
point 7.
152. If from the extremities of any diameter, two chords |
be drawn to any point in an ellipse, and one of them be
91
parallel to a diameter, the other will be parallel to the con-
jugate diameter.
Join any point J (fig. 47) with the extremities of any
diameter PG; and let 2’, y’, be co-ordinates of P referred to
the Axes of the ellipse, and consequently — a’, — y’, those of
G, and x, y, those of J; then if m, m’, be the tangents of the
angles which JP, IG, make respectively with the axis of a,
dered yY+y alt Raine 09
m=——, m' = oom = as
i a av + “ 0 ha — v7
» Oo fo ‘¢
but ay? +a’ =a°b’, ay? + Ba”? = ab’,
9 °
° b ‘ ; 2 I2 b 2 2 Ps erm b :
“., subtracting, y° — y rr rl —@*), .. mm = — —;
which shews (Art. 140) that if GZ be parallel to a diameter,
PI will be parallel to the conjugate diameter. Chords joining
any point in the ellipse with the extremities of a diameter,
are called supplemental chords of that diameter.
Conversely, if two chords, whose equations referred to
the Axes are y=muv+e, y=mae+ec, satisfy the condition
9°
~
mm = — a? then, whether they both pass through the same
point in the ellipse, or through the extremities of a diameter,
they are supplemental to one another.
153. The angles between the supplemental chords of any
diameter whatever are the same as those between the supple-
mental chords of the major axis.
For if from the extremities of the major axis 4, 4’, lines
be drawn parallel to PI, GJ; as their equations will satisfy
>
7
the condition mm’ = — —, they will intersect in a point K
a
in the ellipse, and the angle 4K'A’ = PIG; that is, no angle
can be contained by the chords of PG, but chords relative
to AA’ can be drawn, containing an equal angle. Hence
the angle between the supplemental chords of every diameter
92
} a
will be greater than a right angle and less than 2 tan~ 5?!
these being the limits of the angle contained by supplemental
chords Plane to the axis major.
}
For let w, y, be co-ordinates of the point K referred |
q a
to the axes, then tan K Aw = pies tan KAle = —2— :
C— a v + a
‘ 2at 2ay 2a b°
.. tan AKA = —— aN me ee
prenatal te y (a — 6")
Hence 4 AKA’ is always obtuse; and it is least, viz. a
a
right angle when y=0; and greatest and = 2tan7!—, when}
o o rg oO b
y = 8.
154. Hence we can readily construct two conjugate |
diameters containing a given angle y. Upon the major axis
of the ellipse describe a segment of a circle (fig. 78) con-
taining the given angle +; then because + lies 4
a
and 2tan™? 5
between 17 |
, there is a pair of supplemental chords which |
contain this angle, and therefore the circle must intersect
the ellipse in one, and therefore in two points K, K’; and |
if AK, AK be joined and CD, CP, be drawn parallel to |
them, then CP, CD, are conjugate diameters, and they
include 2 DCP = AKA =+.
a
If the given angle be 2 tan7! ;
the circle will touch the ellipse at B and only one system |
77)
of conjugate diameters will be determined, viz. the equal ones ; |
if the given angle be a right angle the chords 4’K, AK
will coincide with the tangent at A and with 44’,
the axes of the ellipse will be determined.
155.
two lines be drawn parallel to two given straight lines to
and |
If from any point within or without an ellipse |
meet the curve, the rectangles of ae segments will be to |
one another in an invariable ratio.
93
Let O (fig. 49) be the given point with co-ordinates h, k.
Then taking O for the pole and measuring 0 from a line
parallel to the axis major the polar equation to the ellipse
will be (Art. 14)
a’(r sin @ +k)? + b’(r cos @ +h)? = a? B’,
which is of the form 7? + Mr— N= 0, where
a’ bh? — a’k? — 0h? eS
eric ys, ae (70 aes ere ee
a” sin’@ + 6? cos’@
if these be the two values of r.
Now let Pp, Qq, be drawn parallel to CP’, CQ’, which
make given angles a, 3, with Caz; then
PO x Op: QO x Og :: a’ sin’ B +b’ cos’B : a? sin’a + 6 cos’ a
aC eG Wee Artal co),
a ratio independent of the position of the point O.
Hence if we suppose Pp, Qq, to move parallel to them-
selves till they become tangents to the ellipse at points P
and Q respectively, and intersect in a point O without the
curve, we have
OPT OOR 7 CPi -2CQ.
156. To find the area of the ellipse.
Let APQRTA’ (fig. 50) be any polygon inscribed in the
ellipse, and let the ordinates PN, QM, &c. be produced to
| meet the circle on the major axis in p, q, 7, &c. and join
Ap, pq, &c.
Then area of trapezium PNMQ= 4 (PN + QM).NM
b
a
b
=t-(pN+qM).NM= ie trapezium pNM q,
or
wy)
area of trapezium PM 5b
a
area of trapezium pM
94
and since the same ratio exists between every two corre-
sponding trapeziums,
area of polygon APQA’ b
area of polygon ApqA’ a’
and this is true however much the number of the sides of
the polygons be increased ; therefore, supposing the number
to be infinite, in which case the ratio of the polygons be-
comes that of the semi-ellipse and semicircle,
area of semi-ellipse b
area of semicircle a
“4
area of ellipse = - 7a? = rab.
a
Likewise if K (fig. 51) be any point in the axis, and |
QPN be an ordinate to the circle and ellipse, then :
. e b e |
elliptic area ANP =-. circular area 4N Q,
a |
, . DY ae
and triangle PK N = —.triangle QKN;
a
therefore, subtracting,
e . . 5 . Fd
the elliptic sectorial area AK P=-—.circular area AKQ.
a
Also, if a’, 0’, be two semi-conjugate diameters and vy the
angle between them, the area of the ellipse = 7a’b’ sin y
(Art. 145).
And the area of the sector bounded by the semi-diameters—
ss ira b’ sin y:
SECTION VIII
ON THE HYPERBOLA.
Various Forms of the Equation to the Hyperbola.
157. To find the equation to the Hyperbola.
The hyperbola is the locus of a point, whose distance
from a given point is always greater than its distance from
a given fixed line, in a constant ratio.
Let KK’ (fig. 52) be the given fixed line, and §' the given
point, from which draw SX perpendicular to KK’. Let P be
a point in the hyperbola on either side of KK’; and from P
draw PM perpendicular to AK’, and join SP, and let the con-
stant ratio of SP to PM bee : 1, e being greater than 1.
Divide the given distance SX in A so that SA=e. AX,
then 4 is a point in the curve; and assuming 4S’ =p, we
Rave AY =". Through 4 draw Ay parallel to AK’, and
e
take A for the origin, and 4w#, Ay, for the co-ordinate axes,
and let AN=a, NP=y, be the co-ordinates of P; then
SP? = e’. PM*,
or SN?+ NP? =e. NX”,
or (w-p) +y7=e. ( + v) = (p+ ea)’;
- YY =2p(e+1)v+ (e?—- 1) 2’,
~
Dp
or, if we replace the known quantity by a,
e—l1
y= (e?— 1) Qaw + 2”),
the required equation.
96
158. ‘To determine the points where the curve cuts the
axis of #, make y=0, then # =0, or x = — 2a; the value
«=0 gives the point A already known; the other value
— # =2a = AA’, determines the point 4’. Bisect 44’ in C,
then in the equation to the hyperbola making # = —a, we
getty = — (e —1)@, or y= +aVe—1.\/—1, which
are imaginary values; hence the curve does not, as in the
case of the ellipse, meet the line BB’ drawn perpendicular
to AA’ through its middle point; if however we put
WANs Cal b, and take BC, B’C, each equal to 6, BB
will be denoted by 26; and the equation will become
b *
Us ~\/2ae + aX”.
a
159. In order to transfer the origin to C, we must
change # into # —a, since AN=CN-CA;
4]
BP Dips
Y= a {2a (a — a) + (@ — a) = ‘ (uw? — a’).
This form of the equation shews that the origin is the
centre of the hyperbola, and that the co-ordinate axes are
Axes of the hyperbola (Art. 64); but the term, axis, is
more particularly appropriated to the portions of those lines,
AA’ = 2a, BB’ =2b; the former of which meets the hyper-
bola and is called the fransverse axis, and its extremities
are called the vertices of the hyperbola; and the latter,
although the line along which it is measured does not meet
the. curve, is taken for the second axis of the hyperbola,
and is called the conjugate axis. Since b= a\/e? — 1, where
e may have any value greater than 1, b may be either greater
or less than a.
160. To trace the hyperbola by means of its equation.
The equation to the hyperbola referred to its axes is
ie
y=t-J/e -a';
a
hence for all values of « between +a and —a, y is ima-
ginary, and therefore no part of the curve lies in the space
97
bounded by two indefinite lines through 4, 4’, parallel
to BC. When w =a, y=0, and as wz increases positively
from a to co, the two values of y are real and increase
from zero to o, and give the infinite branch ZAz situated
symmetrically with respect to Ca; and since when the sion
of w is changed, the values of y do not alter, the negative
values of w will give a branch Z’d’s" precisely similar to
the former, on the other side of BB’, which is described by
taking SP’: P'M ase: 1. Moreover the two opposite
branches of which the hyperbola is composed, will every-
where turn their convexities towards the axis BB’; other-
wise a straight line might intersect them in more than two
points, which is impossible, (as will appear Art. 186).
161. Since the hyperbola is symmetrically situated with
respect to its axes, if we take CH = CS, CX’ = CX, and
draw kX” parallel to CB, the curve may be described by
means of the focus H and directrix kX’, exactly in the
same way as by means of § and KX. Hence the hyper-
bola has two foci, situated in the transverse axis at equal
distances from its centre.
Also since a = P a we have 4S = a (e —1);
e-—
SOU
“ SC = AC+ AS =a+a(e-1) = ae, and @ see
The quantity e, which expresses the ratio of the distance
between either focus and the centre, to the semi-transverse axis,
is called the eccentricity. Since b = ar/e?—1, the eccen-
2 2"
ticity, e, expressed by the semi-axes, is equal to VG
a
Hence SC =/a? +b, and AS’. A’S' = BC’.
162. Since AS =¢.4N (Art.157), wehave LY = ACE
e
a a CA?
Ah A= -AX =— = —='__,
° 4G Eire ae ES
>
Fa
SS ee eS ee == = =: —
sm -" < Te ag PLA. = Ta —— werL — - = ~ = ~ aaa —
is a ee eee a
98
1 t ale o1) VER |
and SX = AS +AX = ——— = Tas |
which determine the directrix relative to the centre, and)
focus.
The double ordinate passing through the focus is called!
the latus rectum. ‘To find its falue make w= CS = ae
(fig. 53) in the equation to the hyperbola, |
ys Fs (a?e? — a’) = b? (e® —1) = a (Art. 158)
i 20 ,
.yo=t—, « LL’ = 5 OF. ='2a(e— 2).
a a
i
163. If the distance C'S between the focus and centre
be supposed to become infinite, the distance AS’ between
the focus and vertex remaining finite, the hyperbola will
be changed into a parabola. |
; b* |
Since —, = e? —1, the equation reckoned from the vertex’
a”
may be written
9
y = 2a(e?—-1)x+(e—1) a’,
or y = 2p(e+1)7+(e—1) 2’, if AS =p.
SC SC 1
But Carper aca = ee lett SC=0,..€=1]5
AYE:
*, y’ =4pa, the equation to a parabola.
164. Hence the equation y? =2p(1+e)a—(1 —e’) a
is that to an ellipse, parabola, or hyperbola, according aj
Peete, 60F. >is van aie every conic section maj
be represented by the equation y? =ma#+mna’*; and it wil
be a hyperbola, ellipse, or parabola, according as 2 1)
positive, negative, or zero. This is the emia form o
the equation by which the Conic Sections can be collectively
represented.
99
When 6 =a, or e? = 2, the above equations to the hy-
perbola become (Arts. 158, 159)
yan —a, y= 2ar+2",
The hyperbola in this case is called rectangular, and it
is to the ordinary hyperbola what the circle is to the ellipse.
2 9
~
165. Since y” = — (a —a’) = = (#4 a) (w - a),
a a
gives PN* = —_./A'N.AN,
we see that the square of the ordinate varies as the rect-
angle of the distances of its foot from the extremities of
the transverse axis.
166. The equation to the hyperbola results from that
to the ellipse, by changing 6° into —b’, or 6 into b\/—1.
This remark may be of use in enabling us to foresee those
properties of the hyperbola which are analogous to proper-
ties of the ellipse. In general, if any result in terms of its
axes be obtained for the ellipse, the corresponding result for
the hyperbola may be deduced by writing b / —1 for b:
167. To express the distances of any point in the hy-
perbola from the foci, in terms of its abscissa.
By Definition, (fig. 52)
SP=e.PM=e.XN=e.(CN-CX)
ley: (« = =) =ex—a; (Art. 162),
now e€ is greater than 1, and as long as P is in that
branch of the hyperbola of which S is the interior focus,
@ is greater than a; therefore this expression for §'P is always
positive.
HP =e. Pk=e.(CN+CX')=e(#+")=ev+a
€
7—2
—_—
Ne
a ne
——_
Se
Sete
|
100 |
|
168. Exactly in the same way as for the ellipse, it may +
be shewn that the foci are the only points whose distances »
from every point in the curve, can be expressed rationally in |
terms of the abscissa of the point. (Art. 116).
169. Hence, subtracting, :
HP — SP =2a; |
or the difference of the focal distances of any point in the |
hyperbola is constant, and equal to the transverse Axis.
Also the excess of the greater above the smaller focal
distance of a point not in the hyperbola, will be > or <2a,
according as it is situated on the concave or convex side of
the curve.
In SP produced take a point Q on the same side of the.
conjugate axis as S' is, and join HQ (fig. 56), |
then HQ< HP + PQ, SQ=SP + PQ, :
and HQ is greater than SQ; :
. HQ —-SQAP, SQ’+ QVP=SP;
. HQ - SQ> HP - SP>2a.
This property affords a simple method of determining any
number of points in a hyperbola of which we know the
transverse axis and foci. In d’A (fig. 53) produced take any
point #’, and with centres S’ and H and radii respectively
equal to AF, A’F, describe circles intersecting in P, P’;
these are manifestly points in the hyperbola. :
The curve may be described by a continuous motion if
we have a rule HM moveable about the focus H, and a
string SPM fastened to M and to the other focus §, of such
a length that HM — SPM = AA’; then as HM revolves about:
H, if a point P slide along HM so as always to confine a’
101
portion of string PM against it, the point will trace out a
portion of the hyperbola; for we shall always have
HP—- SP = HP+PM-(SP+ PM)=HM-SPM= Ad’.
This property also furnishes the following method of
investigating the equation to the hyperbola.
170. To find the locus of a point the difference of
whose distances from two fixed points is constant.
Through the two fixed points S, H, (fig. 53) draw the
indefinite line Ha, bisect SH in C, and through C draw
Cy perpendicular to it; and take Cw, Cy, for the axes of
the co-ordinates, as the locus will evidently be symmetrical
with respect to these lines. Let SC =CH=c, CN=a,
NP = y, the co-ordinates of any point P, and HP — SP = 2a.
Then HP? =(CN + CH)’ + NP? =(a# +c)? +4’,
SP? = (CN —- CS)? + NP? = (# -c)’ +’;
*. HP® — SP’ or 2a(HP + SP) = 4c2;
2Cé
Nahe ge bt Kay eel
a
but HP —- SP =2a;
CH
» HP=—+4a;
a
CH 2
(= +a) =(@t+ec)’+ y’,
2
cx” Cc’ — a’
4 4M ee: 2 2 2. 9
+ 260040 =a +2060 +0 +y', or y= 2 a —¢? 4 a,
o
+
o
oe ke
or y=
— (a — a’).
=" (a? = at)
Now HP — SP is less than SH, or a es S08 Pr Bee = 22 2
Swaess SS S See e SS SSS SSS
102
171. To find the polar equation to the hyperbola, one |
of the foci being the pole.
Let the interior focus be the pole, and let the polar co-
ordinates of any point P be SP=7r, ZvSP=6 (fig. 52), |
then SN =r cos@, and XN = XS + SN = hla +7 cos Os
e
r=e.AN =a(e?—1) + ercos 8,
or 7(1 — ecos 0) = a(e? — 1),
a(e’ — 1)
1—ecos@>
aks ‘de =
Since e>1, there is some angle «SD =a, (fig. 53) whose |
1 : ;
cosine =—; for values of @ less than a, 7 is negative, and |
e |
there are no points in the branch 4Z corresponding to thosell
values, because 'P = ex — a is always positive. When @=a, |
y is infinite and the radius vector meets the curve at an
infinite distance; when 9 exceeds a, 7 is positive, and as @ |
increases to a we get the portion of the curve ZA; when |
@ increases beyond z, the same values of 1 recur in an |
inverse order, giving the portion Ag, till @ = 27 -— a, when |
r is again infinite and afterwards becomes negative.
172. If § be the exterior focus, and the co-ordinates of |
any point P’ be SP’ =7, ZaSP' = 0, then SN’ = — rcos8@, |
and YN’ = SN’ —-SX = ~ 700s 0 — — (e — 1),
e
“ r=e.XN’ =—recosé — a(e’ — 1),
; — a(e* —1)
or r =—_—_—___.
1+ ecos@
In this case also, r is negative and therefore has no point |
corresponding to it, till @ =m —a, when it becomes infinite, |
and then produces the branch of the hyperbola Z’ A's’ as @-
changes from w-—a to r+a.
There is no difficulty in shewing that if we remove the
restriction of having 7 positive, and measure negative values
103
upon the radius vector produced backwards, the same equa-
tion will represent both branches of the hyperbola.
173. To find the polar equation to the hyperbola, the
centre being the pole.
Let CP =r, 227CP=6 (fig. 53), be the polar co-ordinates
of a point P, whose rectangular co-ordinates are w and y;
then w = rcos@, y = 7 sin@, and substituting in the equation
a’y? — b?a’ = — a’b’, we get
r’ (a’ sin? 0 — 8? cos’ 0) = — ab’,
or, dividing by a’ and observing that — =e —-1,
r’ (1 —e’ cos’ 0) = — Bb’;
b
e hes Le
: 1
Taking @ from zero to a, where cosa =~—, we get the part
e
AZ; from 0=a to 0=7— I® = » Laue Pe = ete eee a
= = Ss < = aes = < =e = = _—— =
ee
=
106
i"
Also since GPT’, GPt, are right angles, and SPG = h PG;
cto DL Lee eened Tad
or the focal distances make equal angles with the tangent
on opposite sides of it. Hence if an ellipse and hyperbola
have the same foci they will cut one another at right angles 3;
for at the point of intersection the tangent to the hyperbola,
since it bisects the angle between the focal distances, will!
coincide with the normal to the ellipse, and therefore be
perpendicular to the tangent of the ellipse.
181. These properties furnish a simple method of draw-'
ing a tangent to the hyperbola through a given point.
First, let the point be in the curve as P (fig. 55). !
Join SP, HP, make HK =24AC, join SK and draw)
PY perpendicular to it; then in the triangles SPY, KPY3
PK = HP -2AC=SP, PY is common, and the angles at
Y are right angles: .. 2 SPY = KPY, and consequently PY|
is a tangent at P. |
182. Next let the point be on the convex side of the
hyperbola (fig. 80 and 81). Join the proposed point 7’ with the/
more remote focus H, and with centre H and radius = 24C
describe a circle cutting HT or HT produced in O. Then
if Z' falls within the circle, JJ'S is less than J'H, and S$
is necessarily outside the circle; but if 7’ falls without the
circle, HT’ — ST’ <2AC < HT -—OT, and therefore ST > OT;
consequently in both cases a circle described with centre 7’
and radius 7'S' must intersect the former circle in two points K, |
K’. Join HK meeting the hyperbola in P and join 7'P, then |
LP is a tangent at P. For in the triangles 7 PK, TPS, |
SP= HP+2AC=PK, TS =TK, and PT is common; |
.. PT’ bisects the angle SPH and is therefore a tangent at P.
Similarly, if HK’ be joined and produced to meet the hyper-
bola in Q, a second point of contact will be determined. The |
two tangents will belong to the same, or to opposite branches :
of the hyperbola, according as the point in which they in-|
tersect lies in 4 LCI or its opposite, or in 2 LCL’ or its op- |
107
‘posite (Art. 176) ; and the angles which the tangents subtend
at either focus will in the former case be equal to one another,
and in the latter supplementary to one another.
For in fig. 81, it is evident that zTKH=TRK’'H;
. TKP= TK Q,... TSP=TSQ, and THP=THaQ.
Again in fig. 80, 2TKH=TK’'H, «. r- TKH = TK'Q,
or rw —- TSP= T7SQ; and «c—- TAP = THK = THQ.
183. The locus of the feet of the perpendiculars dropped
from the foci upon the tangent to a hyperbola, is the circum-
ference of the circle whose diameter is the transverse axis.
For joinng CY (fig. 55), since S'H is bisected in C,
and SK in Y, CY is parallel to HK and =} HK = AC.
Also, drawing HZ, CQ, perpendicular to ZY, Q is the middle
point of ZY, and therefore CZ = CY = CA.
184. Since C is the centre of the circle which is the
locus of Y and Z, and SYZ is a right angle, if SY and
ZC be produced to meet in S$", this will be a point in the
circumference; and from the equal triangles SCS’, HCZ,
SS’ = HAZ: |
mes rex AZ = SY x. SS = SA SA = BC? (Art. 161),
HZ ST Mee is
RY 4
ince —— = —--~ (Art. 180 Pa =
Also since Sp HP (Ar ), or AZ Hp? We have
SP
“. SY?= BC ;
‘HP
or, if SP, SY, be denoted respectively by r and p,
Ye
| jie aa
2a+r
185. Draw CE parallel to PY (fig. 55), then CP is a
parallelogram, and PE =CY=CA.
All the properties of the ellipse proved in Arts. 132136,
may without difficulty be extended to the hyperbola.
108
The Hyperbola referred to its Conjugate Diameters.
186. To determine the intersection of a straight line’
with a hyperbola. |
Exactly in the same way as for the ellipse, it may be)
shewn that the ordinates of the points of intersection of a
straight line and hyperbola, whose equations are respectively |
y=maer+e ay a2’=—- ad’,
are the roots of the equation
(b° — m? a’) y? — 2b’ cy’ + (c? — ma’) b= 0
Hence the line cannot cut the curve in more than two
points, and if the roots are impossible, it will not meet the |
curve at all. If the roots are equal, the line will touch the |
hyperbola; and we get y=ma2+/ mia — b, for the equation |
to the tangent of the hyperbola, in terms of the angle which |
it makes with the transverse axis.
187. To find the locus of the middle points of a system
of parallel chords. |
Let the chords be parallel to a line CW through the |
centre (fig. 57), whose equation is y= ma; then the equa-
tion to any one of the chords QQ’ will be y=ma-+e, and |
to determine the points in which it meets the hyperbola, we |
must combine its equation with that to the hyperbola,
ay — ba? = — ab:
e e e e e e e 1
this gives, eliminating w by the substitution — (y — c),
m
: 2b'ec 12 ec — ma’ ae
J Tip are ‘B-ma ”
the roots of which will be represented by QM, Q'M’ if the |
values of y and the corresponding values of x, are all positive, -
but if one or both values of wv are negative, the line will meet |
the opposite hyperbola; then if V be the middle point of QQ’,
and CN = X, NV = Y, its co-ordinates, 2NV = QM + Q’'M’;
109
b7¢
a A 79 Se gee
b° — ma’
1 MAC
and A= ee (Y -—c) = sy Maem
m b* — m*a’
Dividing one result by the other, in order to eliminate
the quantity ¢ which particularizes the chord, we get
b°
ma*
WS A,
a relation between the co-ordinates of the middle point of
any chord, and therefore the equation to its locus, which is
consequently a straight line CV passing through the origin.
The straight line which passes through the middle points
of a system of parallel chords is called a diameter; hence
all diameters of a hyperbola pass through its centre; and,
conversely, every line through the centre may be considered
as a diameter.
188. Hence, denoting the equation to any chord QQ’ by
y=mex-+c, and the equation to the diameter CV which bisects
it by y = m'x, we have
2 2
vee, b
ig i Or mm =~ >>
a simple relation, by means of which the equation of one
may be deduced from that of the other.
189. If a diameter CV bisect the chords parallel to
another diameter CW, then likewise the chords which are
parallel to CV are bisected by CW.
For any one of the last-mentioned chords RR’ may be
represented by the equation y = m'’ax +c’; then the diameter
which bisects it will have for its equation
Y= rats OY = me, which belongs to CW.
Se
= >= SS
EE
——
—— ee
110
Hence two diameters, whose equations y = ma, y=m' x, are
2
b fl
so related that mm’ = =? have the property that each bisects’
the chords parallel to the other; they are called Conjugate!
Diameters. But the term is usually restricted to those
portions of them PP’, DD’, which are intercepted by the)
proposed hyperbola, and the hyperbola which is conjugate
to it (fig. 58).
190. The latter is a hyperbola BD B’D', whose trans-
verse and conjugate axes are respectively equal to, and in
the same straight line with, the conjugate and transverse
axes of the proposed curve; and the employment of it is |
attended with great conveniences in stating and investigating |
the properties of the hyperbola.
The equation to the conjugate hyperbola, referred to the |
same axis of w and axis of y as the primitive hyperbola,
so that DM =y, CM =2, (fig. 58) will consequently be
9
a”
oO b” ;
o = = (y’ ~ ), or 9? = - (a? +0’),
which we observe results from the equation to the primitive |
hyperbola (Art. 158), by replacing a* and 6? by — a? and — 2.
191. If PT be a tangent at P (fig. 58), and 2’, y', the.
co-ordinates of P, then the equation to PT’ is
D0 R
Yo rast (v—w); (Art. 174)
/
but the equation to CP is y = a w, and therefore the equa-
c
tion to CD, the diameter conjugate to CP, is
b? x’
ef =
a ee
ay
which represents a line parallel to PT’. Hence the tangent
at the extremity of any diameter, is parallel to the corre-
111
sponding conjugate diameter. Similarly, the tangent to the
conjugate hyperbola at D is parallel to CP; and if tangents
be applied to the hyperbola and its conjugate, at the ex-
tremities of a pair of conjugate diameters, they. will form
a parallelogram inscribed in the two curves, whose sides will
be bisected in the points of contact.
192. Of any two conjugate diameters, only one can
meet the hyperbola.
Let y = mw be the equation to a diameter; to determine
its intersection with the curve, put mw for y in the equation
ay? — Bx? = — a°b’,
and we find for the abscissz of the points of intersection
b
which values are real as long as m is less than —, but imaginary
a
. Oper. |
if m be greater than —; in the former case the diameter
a
intersects the curve, in the latter it does not. But the rela-
; b° : b :
tion mm’ = — shews that if m be less than —, m’ is greater
a a
b
than —; hence every diameter which meets the hyperbola, has
a
its conjugate diameter amongst those which do not meet it.
193. If we construct on the axes of the curve, the
rectangle Ll’ (fig. 59), all the diameters which fall within
the angle ZCl, make with AC an angle whose tangent
A : , b J oe
(abstracting the sign) is less than -; whilst the diameters
a
which fall within the angle LCL’ make with AC an angle
b
whose tangent exceeds —; the former are those that meet
a
the curve, the latter those that do not.
= ——— = = = =
ee ee —— —=
112
b bi
In the particular case when m = —, we have also m’ = 7" e|
x |
and the conjugate diameters coincide with El’; and as the
value of # (Art. 192) becomes infinite, they meet the curve |
: cee b |
only at an infinite distance; similarly, when m= —-—, we
a
b e e 6 e |
have m’ = — —, and the two diameters coincide with the other |
a
diagonal Z’/, and meet the curve only at an infinite distance.
The lines Zl’, L’l, are, for this reason, called Asymptotes ;
and they correspond to the equal conjugate diameters in
the ellipse.
194. Having given the co-ordinates of the extremity of |
any diameter, to find those of the diameter conjugate to it.
Let CD be conjugate to CP (fig. 58), and let it meet
the conjugate hyperbola in D; let 2’, y, be the co-ordinates
of P, and consequently y = 7 @ the equation to CP, thea
&
Bia nat
Y=, 1s the equation to CD; and to determine the
ary
co-ordinates of D we must combine this equation with the
equation to the conjugate hyperbola, which is
ay? — Bu? = ab.
Oa 4
~
This gives, eliminating y by the substitution ay”
w? (Ba?
° ba’
7 = aed — =-— °
w= CM = ~~, and ¢ DM ys
the other pair of values of a and y belonging to the point D’.
113
195. The difference of the squares of any two semi-
conjugate diameters, is equal to the difference of the squares
of the semi-axes.
CP? = x? +. y!®.
6?
/ oO "29 , 2
me = 2° —a° ty”? +B, because y = a w?-a’);
«5 CP? —- CD? = a? — B*.
196. All parallelograms whose sides touch a hyperbola
nd the conjugate hyperbola at the extremities of a pair of
onjugate diameters, are equal to one another.
Draw PF perpendicular to CD (fig. 58), then (Art. 191)
rea of whole parallelogram = 4CD.PF=4CD.CTsin TOF
be’ a?
tie pa 4ab.
a av
4DM.CT =4
Il
197. Draw the diagonal CL (fig. 58), which will pass
rough the middle point of DP, whose co-ordinates are equal
14(CM+CN), 4(DM + PN);
~
*. tan LCA =
hich value is independent of the positions of CP and CD;
nce the parallelograms whose sides touch a hyperbola and
3 conjugate at the extremities of a pair of conjugate dia-
eters, are not only equal in area, but they all have their
agonals in the same line; namely, the diagonal of the
ctangle whose sides are the semi-axes.
198. If we denote CP, CD, by a’, 6’, and z PCD by y;
» have
PF =a siny, and. a’b’ sin y=CDx PF =ab.
3
114
Also if we denote PF by p, we have the relation between)
the central distance of any point and the perpendicular from)
the centre upon the tangent at that point, :
> GS watts eatin
P~ = Fj 9 — ast ake =
CD ta a4 0:
The magnitude and position cf two conjugate diameters)
that include a given angle, may be determined in the same
manner as for the ellipse (Art. 145). |
199. The rectangle contained by the focai distances of|
any point, is equal to the square of the corresponding semt-
conjugate diameter. |
ie =
2_ @4 bP
=e 2 — a’ =(ea +a). (ex — a)
SNe Sra Bet
200. To find the equation to the hyperbola referred ta
the system of oblique axes formed by any pair of conjugate
diameters.
The equation to a hyperbola referred to its centre and
axes 1S
g 279
ay —Pa=— 02
Let the conjugate diameters CP, CD (fig. 60), be the
new axes of a and y’, inclined to the axis of # at angles
PCA =a, DCA =; then since the origin remains unaltered,
the formule for passing from the rectangular to the oblique
axes are (Art. 42) |
, , oe r e
v=wvcosaty cosB, ya’ sinat+y sinp.
Hence, substituting and reducing,
(a? sin? a — b° cos’ a) a”? + (a* sin’ B — b* cos’ 3) y”
9
+ 2a'y' (a? sina sin B — b? cosa cos B) = — a0:
115
>
~
b . . oO
but tan a.tan 8 =— ; therefore a? sin a.sin [3 — b° cos a.cos 3 =0.
Ai
iso ihsGues a. O1).— b. wa hace Arts. 173 and 190
5 3
a” (a’ sin? a — b? cos?.a) = — ab’,
b” (a? sin? B — B cos* 3) = + a?b?;
hence, substituting and dividing by — ab’, we get for the
required equation
7 a he
or, In a geometrical form, supposing PC produced to meet the
curve in G,
CD
pre Bere.
a> Gp
201. This equation, which, suppressing the accents of
the variables, is
a?y? — 7 a? = — gb”,
being of precisely the same form as that relative to the AXES,
it follows that all properties which do not depend upon the
inclination of the co-ordinates, will be common to the axes of
the hyperbola and to its conjugate diameters.
Hence the equation to the tangent at a point Q(a’, y)
will be
, , as ‘
a*yy — xa’ = — ab”,
and if the tangent meet the axis of the abscisse in T', we
OF; :
shall have C7'= cp? 3 before; and if we wish to draw
a tangent through an external point Q(h, k) (fig. 61), we
shall have, to determine the points of contact (a, y’), the
equations
fo Ve Oo ‘
ay? — b? gl? = — g!p!?,
fy Fe fe o
a*ky — b°ha' = —a@?b’;
116
the latter, considering wv and y' as the variables, being the
equation to the chord joining the two points of contact 5
and if we construct this line by taking
a! : Sa b”
OE ro 5 CR = k 3
and joining RT’, it will cut the hyperbola in the two points
of contact.
202. Since the distance CJ’ is independent of k, if
through Q we draw a line parallel to CD, and from any |
other point in this line we draw a pair of tangents to the
hyperbola, the secant passing through the new points of |
contact will cut the diameter CP in TJ, as this point only
changes when # changes. Hence, if from the several points
of any straight line, pairs of tangents be drawn to a hyper-
bola, the straight lines which join the corresponding points
of contact will all intersect in the same point; and conversely
if through any point we draw different chords and apply
two tangents at the extremities of each, the locus of the |
intersection of the tangents will be a straight line.
203. The tangents at the extremities of any chord will |
intersect in the diameter of which the chord is an ordinate.
For, taking that diameter and its conjugate as the axes |
of w and y, the equation to the tangent will be
ta?yy’ — baa = — ab”,
according as we consider the point Q(a’, y’), or the other
extremity of the chord Q’ whose co-ordinates are a’, —y,
to be the point of contact; and in both cases when y =0,
“2
a i :
v= -,; therefore the tangents meet the axis of # in the
¢
same point 7. (fig. 60).
Exactly in the same manner as for the ellipse (Art. 152),
it may be shewn that if from the extremities of any diameter
two chords be drawn to any point in a hyperbola, and one
of them be parallel to a diameter, the other will be paral-
lel to the conjugate diameter.
117
204. If from any point within or without a hyperbola,
two lines be drawn parallel to two given straight lines to
meet the curve, the rectangles of the segments will be’ to each
other in an invariable ratio.
Let O (fig. 62) be the given point with co-ordinates h
and &; then taking O for the pole, and measuring @ from
a line parallel to the transverse axis, the polar equation to
the hyperbola will be
a° (r sin@ + k)* — b (r cos + h)? = — a°b°,
which is of the form 7? + Mr — N= 0,
—@ +h”
where V = —— : —— =rr',
a” sin’ @ — b? cos’ @
if these be the two values of +.
Now let Pp, Qq be drawn parallel to CP’, CQ’ which
make angles a, 3, with Cw, then
PO x Op: QO x 0¢ :: a’sin® B — B cos? B : a® sin? a — b? cos? a
SC La CQ ce (ATt11 79)
which ratio is independent of the position of the point O.
Hence if we suppose Pp, Qq, to move parallel to them-
selves till they become tangents to the hyperbola at points
P and Q respectively, and intersect in a point O outside the
curve, we have
ORs OG a CPE a GOL,
The Hyperbola referred to its Asymptotes.
205. The diameters which never meet the hyperbola
at any finite distance, are called Asymptotes.,
These diameters coincide with the diagonals of the rect-
angle constructed with the semi-axes (Art. 193); and we
shall now shew, according to the strict notion of an asymptote,
that although they never meet the curve they approach in-
definitely near to it. For, the equations to CL (fig. 59),
SS See —— a nee De ana oniee a oe
eee eee a ee ——— ee
——
——
eS
SS
118
and to the hyperbola, when referred to the axes of the |
curve, being, respectively, (Arts. 19 and 159)
b Miya
Yy = Vs q = J x La a’,
a
a
the difference of the ordinates
4 , b a a tee
PP =-(«#- Vier aed ii)
a
therefore, as w increases, this difference continually diminishes,
and ultimately vanishes when «= @. Similarly, it may be
i b
shewn that CL’, whose equation is y = — —a, approaches inde-
3 J 0 5
L
finitely near to the other branch of the hyperbola.
It appears, by Art. 176, that the asymptote to the hyper-
bola, ‘is the limiting position of the tangent, when the point
of contact is infinitely distant.
206. When the hyperbola is referred to a pair of con-
jugate diameters, the directions of its asymptotes will be
determined by the diagonals of the parallelogram constructed
with the diameters; for those diagonals always coincide with
the diagonals of the rectangle constructed with the semi-
axes (Art. 197). Also in this case, where the co-ordinates
are oblique, we may shew, exactly in the same manner as
for rectangular co-ordinates, that the diagonals approach in-
definitely near to the curve. For, the equation to the diagonal
CL (fig. 63), and to the hyperbola, referred to the conjugate
diameters CP, CD, are, respectively,
b’ bs p=
Jaron es y= @ -a 3
a a
a b!
C+ A/ x yon
Hence when «=o, RQ becomes zero; hence Ll’ ap-
proaches indefinitely near to the portions PQ, P’Q’; and
similarly it may be shewn that L’l approaches indefinitely
near to the other portions of the curve.
b! ee
~ RQ=—(#- V/ ae — A ees
a
119
207. If any chord of a hyperbola be produced to meet
the asymptotes, the parts of it intercepted between the curve
and the asymptotes will be equal.
Let Qq (fig. 63), any chord, when produced cut the
_asymptotes in R, 7; bisect Qq in J, join CV cutting the
hyperbola in P, and refer the hyperbola to the diameter CP
and its conjugate CD; then the equations to CR, Cr, are
|
| b’ b’
| deen ap Se ke
VR =Vr, and VQ= Vq; .:. subtracting QR = qr.
| Also RQ. Qr = (RV + VQ) (RV — VQ) = RV? - QV?
= =, ja? - (a - a’) = 6? = CD’;
i.e. the rectangle of the segments into which a line, ter-
minated by the asymptotes, is divided by the curve, is equal
to the square of the semi-diameter to which it is parallel.
If a line be drawn through P parallel to Rr, it will
be a tangent at P, and PL=Pi. Hence every tangent
‘terminated by the asymptotes is bisected in the point of
‘contact.
208. rom any point P (fig. 63) of the hyperbola, draw
parallels PG, PF to the asymptotes, and draw the tangent
‘Li which is bisected in P. Then the parallelogram GY is
half of the triangle LC1.
But area LCI is constant, whatever be the position of
P, and equals ab (Art. 196).
Hence, denoting by 2a the angle between the asymptotes,
and by «, y, the co-ordinates of P referred to the asymptotes
as axes, so that Cf =x, FP =y, we get
ab ; 2 tana 2ab
vy sin2a = ee but sin2a = —
1 +tan’a a? +O?’
er Se
Se
es
=
iy W
Wd
since tana =-;
a
“ vy=1(a +b’),
the equation to the hyperbola referred to its asymptotes.
which may be likewise obtained in the following manner.
209. ‘To transform the equation to the hyperbola referreé
to its axes, into that representing the hyperbola referred td
its asymptotes. |
Let the inferior asymptote be the axis of wv’; and lef
CM =a, MP=y’, (fig. 59) be the co-ordinates of a point
referred to the asymptotes, and CN =a, NP=y, the co.
ordinates of the same point referred to the axes of the
hyperbola; and zLCA=1C4A=a;; then drawing MQ, MR.
respectively perpendicular to CN, and to PN produced, we have
w= NQ+ QC =y' cosa+wz’ cosa,
y=PR-RN=y sina-~w' sina;
hence, substituting in the equation a?y? — b?a = — a?b®, we get!
a’ sin’ a (y’ — a’)? — B cos’ a (y’ + a’)? = — ab?
b?
b
but tana =-; and .-. a?sin’a = B’cos?q =
If a=6b, tana=1, 2a= ta; and the asymptotes are
then perpendicular to one another. The hyperbola with
equal axes is therefore called rectangular, and its equation |
is LY = da’.
210. To find the equation to the line touching the hyper- |
bola at a given point, when referred to its asymptotes. |
Let the co-ordinates of the given point be a’, y’, and those |
of a point near it a”, y”; the equation to the line passing
through them will be
fy
I re}
c u
© — 2’),
but ay =1L(a°+8%), aly =t@+6);
ps ay” ae a! y! = 0, or a’ (y” af y’) fe y (ar oe x’) =y :
” ,
y-y sy
. ” ye vA
av ee: av v
Hence the equation to the secant becomes
i bea eet (owe w) 5
* and in order that it may become a tangent, we must suppose
‘yy =y', which gives, for the equation to the tangent,
/
?
y :
y-y=-F@—a), or ay tya=2aly = 4 (0° +d’).
To find where the tangent cuts the asymptotes, make
y=0; .«. 2 =2e, or Cl=2CF, and Li=2LP (fig. 63),
agreeably to Art. 207.
211. To find the area PNMQ contained between a hyper-
bola, its asymptote, and two ordinates to the asymptote.
Let the equation to the hyperbola be wy=a’*, and
LyAuv=w (fig. 76), AN=a, AM=6; take a such that
w\” 6b & “Tb | é ‘
—~| =-,or-= —,and take the abscissze in geometri-
a a a a
cal progression, so that
2 3 n
AN, = 2; AN, =; AN, = =» &e 4M =
and complete the (7) parallelograms PN, P,N2, &e.; then
; v@ :
as m increases, — tends continually to 1, and therefore the
a
difference between any two consecutive abscisse continually
diminishes; and consequently the limit of the sum of the
parallelograms, when is infinite, is the hyperbolic area
PNMQ.
122
PN, = (# — a) asing,
2
a
— &@
But area of parallelogram
2
av
a
area of P,N,
—sinw = (# — a) ASIN w,
v
a?
area of P.N;
Il
(= =)
hee 741
—sinw= (wt —a)asing,
2
ereceewe eee OSCe eee eee eee veeeseeeee
therefore the sum of the parallelograms = n (a — a) a sin w
if
by” |
=«°sina.n} (7) if ;
a
]
.. hyperbolic area PNM Q =a? sin w. limit 2 (") as VC = 6. ) |
a
1
ous Gite ) a
a’ sin w. limit » {(1+ 5 1)*— 1} |
ye b 1 Wa b
a” sin w [= +1 (- - 1) eat
a ONG a
© e b
AGS 4
a
Mee b |
a~ sin w log {-}.
a
If w= ta, and a=1, then area PNMQ = log AM ;
or the area is the logarithm of its. abscissa; on this account,
the Napierian logarithms are sometimes called Hyperbolic.
SECTION IX.
ON THE SECTIONS OF THE CONE AND CYLINDER,
912. Tue surface described by an indefinite straight line
which is carried round the perimeter of a given circle, always
passing through a fixed point, is called a cone (fig. 64).
The circle is called the base of the cone, and the fixed
point its vertex, and the line joining the vertex and centre
of the base is called the axis. ‘The cone is moreover right
or oblique, according as the axis is at right angles, or
inclined, to the plane of the base.
As the generating line is unlimited in both directions
from the vertex, the surface of. the cone is. composed of
two portions or sheets, perfectly similar, situated on opposite
sides of the vertex. Also from the mode of generation it
follows that every plane parallel to the base will cut the
cone in a circle; and every plane through the axis will cut
it in two straight lines. When the surface is a right cone,
every generating line will make the same angle with the axis.
The different curves obtained by cutting a cone by a
plane are called Conic Sections.
Sections of a Right Cone by a Plane.
213. All sections of a right cone made by a plane are
curves of the second order.
Let PAP’ (fig. 65) be a section of a right cone made
by any plane; and through the axis VO draw 4q plane per-
pendicular to that of the section, cutting the cone in the
lines VB, VD, and the plane of the section in the line
AN, which take for the axis of w Through any point P
of the curve AP draw a plane perpendicular to the axis,
intersecting the cone in the circle MPQ, and the plane of
124
the section in PP’; then MQ will be a diameter of the’
circle, and PN will be at right angles to both AN and NM,
and will consequently be a common ordinate to the circle
and conic section AP. Draw Ay parallel to PN and take.
it for the axis of y, and let AN=a, PN=y be rectangular
co-ordinates of P; and choosing the data so as to embrace|
every case, and therefore not assuming that 4N meets VQ.
produced, let AV=d, zVAN=0, AVQ=2a. Then since
PN is perpendicular to the diameter MQ,
|
y= MN x NQ. |
M sin @ xv sin 0
But WIN arcosq ed tt Nee cosa
And drawing NF' parallel to QV,
since 4 ANF = GFN—GAN = VAG — (VAN - VAG)
=2VAG-VAN=7 - 2a -8,
AF _sin@a+6) E AP A250 Ga + 9) |
AN COS cud eh cosa ‘
#.sin(2a+0)_
<<). sabes e+. Wwe
COS a
seh 18 0 sini
2 2d.sina.sin@ sin@.sin(@Qa+0) ,
: —_——__.____~ yx’
. Y =——__ r-
2 2
COS a@ COS" a
the equation to a curve of the second order; therefore every
conic section is a curve of the second order ; and it will be an
ellipse, hyperbola, or parabola, according as the second term
is negative, or positive, or zero, (Art. 164).
Now the second term can only change its sign when
sin (2a +@) changes its sign. Hence the section will be an |
ellipse as long as 2a + is less than , and therefore 4N
meets VQ produced, or the cutting plane meets only one
sheet of the cone.
It will be a hyperbola when 2a +60 is greater than 7,
and therefore 4N and VQ intersect when produced back-
wards, and the cutting plane meets both the sheets of the
cone.
|
125
It will be a parabola when 2a+0=7, and therefore
4N, VQ are parallel, or the cutting plane is parallel to a
yenerating line of the cone.
7
214. To determine the axes of the conic section, we
nave, since the co-ordinates are rectangular, by comparing
she equation (supposing it to represent an ellipse and there-
fore sin (2a +6) to be positive)
P 2dsina.sin@ sin 0. sin (2a +9)
tT) cosa cos’ a
ob? b?
with y? = —av- —2’,
a a’
2b* ;
the latus rectum or — = 2d. tana. sin@,
a
Bb? sin@.sin (2a + 8)
and — = ——_—______—“;
a’ cos’ a
2dsina.cosa
sin (2a+ 0) °
2d’ sin® a sin 0 : sin 8
-. 26° = —__—_____,, orb=d.sina Ne ;
sin (2a + 8) sin (2a + @)
Bi AAT fig
215. The minor axis may however be more conveniently
expressed in the following manner.
From the extremities of the axis major let fall perpen-
diculars AF = f, A’G = g (fig. 64), upon the axis of the cone ;
and through C, the middle point of Ad’, draw a plane
parallel to the base, cutting the section in BB’ which is
its minor axis, and the cone in the circle MBQ; then
BC = MC x CQ=A'G x AF = fe,
because MC, being parallel to DA’, = 4 DA! = AG
and similarly CQ = AF.
126
Hence the distance of the foci of the elliptic section = AD :
for, dropping the perpendicular 4K, 4’E =f +e |
| AD = 4a" + 4g" — 4g (f+ g) = 40° — 4fg = 4 (a? — 2);
AD= ar/ a? — 6 = distance of foci.
216. If in that. section of a cone through the axis
which is perpendicular to the plane of an elliptic. section, we!
describe circles touching the generating lines of the cone and |
the axis of the section, the points of contact with the axis
will be the foci of the section.
For the distance of the foci = d'D’ (fig. 66).
But AD’ = 4U' -D'U'=AS—AU
= AA’-2AS8;
AS = A ade
therefore § is a focus. Similarly, H may be shewn to be
the other focus.
Produce UU" to meet AA’ produced in LY,
then. from the similar triangles AUX, ADA’,
ube AU | ANGmesse
AA AD 2dC 25C°
AX AS Ca AC
; = or -
AG! SC MG Se
therefore X is the point where the directrix meets the axis.
(Art. 110). Similarly, X’ is the point where the other directrix |
meets the axis.
217. When the section is a hyperbola, the equation is
2
a =
2dsin@. sin la sin in 8. sin (a+ ABE
L—
cos a ~ cos? a
127
where sin(2a@ + @) is a negative quantity, and consequently:
the second term is positive; by comparing this with
Mek CAs ie
Ye = —_ W# + ae, “a
a a
we may determine the axes of the curve, as in the case of. the
ellipse. When in this case d = 0, the equation becomes
sin @.sin (2a + 6)
one v
9
r)
y=
cos” a
which represents two generating lines of the cone.
In this case also, the semi-conjugate axis is a mean pro-
portional between the perpendiculars dropped from the vertices
of the hyperbola upon the axis of the cone; and the distance
of its foci is equal to the portion of the slant side inter-
cepted by the perpendiculars.
d’ sin* a sin @ d sin @
For 6? =—___—_——. = d sin’ a. .
— sin(2a + @) sin (27 — 2a — 8)
=AV.AV.sin’a'= AF. A’G'= fe (fig. 82);
and AD? = 4a? + 49° —4¢(g—f) = 40° + 4fg = 4(a + 6’).
218. When the section is a parabola, or 2a+0=7
the equation is, since sin@ = sin 2a,
by;
2dsin@.sina a9
yo er A I"
cosa
219. We must now demonstrate the converse proposi-
tion, namely, that curves of the second order are conic
sections.
Every curve of the second order is contained in the
equation
y = 4px + ne",
where 4p is the latus rectum, and » the square of the ratio
of the axes, abstracting the sign. What we have to demon-
strate is, that the quantities p, », and a being given, we
128
can assign real values of d and @ which shall render the
above equation identical with
: 2dsina.sin@
y=
sin@.sin(2a+0) ,
ih — Mh =? a a~
cosa cos’ a
Equating the coefficients of a and # in the two equa-
tions, we get ‘
dsina.sin@ p sin 9 .sin (2a + 0)
— «Ps» 5 = = Ns
COS a COS" Qa
the former of which will give a real value of d when 0 is
real; the latter may be transformed into
4 $cos 2a — cos (2a +260)} = — ncos’a,
or cos2(a + 0) = 2(1 + 2) cos’a — 1.
In the ellipse, 2 is negative and less than 1; hence the
preceding value of cos2 (a+) lies between +1 and -1,
and therefore @ is always real ; consequently any given ellipse
may be regarded as a section of any proposed right cone
whatever.
In the hyperbola, m is positive and of any magnitude ;
if the above value of cos2(a+@) be negative, it will be
evidently less than 1, and 9 will be real; but if it be
positive, we must have, in order that @ may be real,
2 (1 + 2) cos’a — 1 less than 1,
I a
and .. cosa less than ————., or than Mey a
V1l+n’ V/ a? +B
but if w be the angle which the asymptote makes with the
a
transverse AXIS, COS wm = » «+» COSaAw 3
/ a? +b?’
and therefore, in order that a given hyperbola may be cut
from a given cone, the vertical angle of the cone must be
not less than the angle between the asymptotes.
In the parabola, 2=0: therefore sin @=0, or sin (2a+0)=0;
the first is inadmissible, for it makes p=0; the second gives
2
129
2a+0= 7, which will always furnish a real value for 0;
hence a given parabola may be cut from any proposed cone
Sections of Cylinder and Oblique Cone by a Plane.
| 220. To determine the curve which results from the
| intersection of a right cylinder with a plane.
Let APA" (fig. 67) be a section of a right cylinder,
AAD a section of the cylinder through its axis, perpen-
dicular to the plane of the section. Through any point P
draw a plane perpendicular to the axis of the cylinder,
intersecting it in a circle whose diameter js MQ, and the
plane of the section in PP’ which will be perpendicular to
MQ, AA’, and will be a common ordinate of the section
and circle.
Let AN=2x, NP=y, AA’ = 2a, AD = 2r,
then y= MN.NQ:
MN AN y
but AD = AA”? or MN = Pipe
NQ NA 7
AD da oe NE Aa a)
A
of
—¥= 2 (2ax — x’),
she equation to an ellipse.
221. In the same manner the nature of the sections
of an oblique cone may be determined; but this, as well
is the discussion of the sections of Conoids or figures gene-
ated by the revolution of conic sections about their axes,
nay be more conveniently deferred to Geometry of Three
Jimensions. There is however one important property of the
blique cone which admits of a simple demonstration, viz.
hat it may be cut by other planes besides those parallel
s 9
130 |
to its base, so that the sections may be circles, and whiclt |
we shall give here. |
Let VBD (fig. 68) be the principal section of an oblique |
cone, that is, a section made by a plane through its axis |
perpendicular to its base; and let MPQ, APA’, be two
sections made by planes perpendicular to BVD, and of which
the former is parallel to the base, and is therefore a circle |
with diameter MQ; and as PN is perpendicular to MQ, |
we have PN? = MN.NQ; and the latter will also be a)
circle, if Zz AA'V = ABD; for in that case the triangles:
NA MN
AMN, A'NQ are similar, and We 7 WA?
»- AN.NA = MN.QN = PN’,
and as PN is perpendicular to AA’, the section APA’, which’
is called a subcontrary section, is a circle; and it is deter-
mined by two conditions (1) its plane is perpendicular to)
the principal section of the cone, and (2) its plane makes!
the same angle with one of the generating lines of the cone
which are in the principal section, as the plane of the base}
does with the other. |
SECTION xX.
ON THE GENERAL EQUATION OF CURVES OF THE SECOND ORDER,
AND ON CERTAIN GENERAL PROPERTIES OF ALGEBRAICAL
CURVES.
Reduction of the General Equation of the Second Order.
222. WE shall now proceed to the reduction of the
general equation of the second degree
ay’ + bay + ca’? + dy + ex + f= 0,
where we suppose the co-ordinates rectangular ; for if they
were oblique, by transforming them to rectangular co-ordi-
nates we should obtain an equation of the same degree as
the above, and which could not therefore be more general
than the one we have assumed. We shall prove, as affirmed
at Art. 62, that this equation by giving a proper position
and direction to the origin and axes of the co-ordinates,
can always be reduced to one of the forms,
Ay’ + Ba’ =C,
2
y = Aa,
‘the co-ordinates being rectangular; and therefore can never
‘Tepresent any other curve than one of those discussed in
the preceding Sections. The principle of the method is to
change the system of co-ordinates, without giving any par-
ticular values to the quantities which determine the position
of the new axes. By that means, indeterminate quantities
are introduced into the transformed equation, to which such
values can afterwards be assigned as will destroy certain of
its terms. Instead of altering both the origin and direction
of the co-ordinate axes at once, it is more convenient to
effect these changes separately, in the following manner.
9-—2
132
923. The general equation of the second order being
ay’ +bay+ca°+dy+eant+f= (a, y) = 9
in order to get rid of the terms involving the simple powers
of w and y, we must change the origin without altering the
direction of the axes, by putting (Art. 39) «=a +h, y= y +k;
this gives
ay? + ba'y + cn + (2ak + bh + dy! + (Ach + bk +e) a
tak? +bkh+ch? +dk+eh+f=09,
and equating the coefficients of wz and y’ to zero, we get
2ak+bh+d=0
semper (hy
Qch + bk +e=0
which give for the co-ordinates of the new origin, provided |
b? —4ac be different from zero, the single pair of deter-
minate values
2ae—bd ke 2cd — be
b? —4ac 6? — 4ac
Hence the equation becomes, suppressing the accents,
ay’ + bay + ca’ + ph, k) = 0,
d’ — bed + ae”
b*? —4ac
>
where @(h, k) =f +4 (dk + eh) =f + :
as appears by multiplying equations (1) by & and h re- |
spectively, and taking their sum; and since this equation
remains unaltered when we change w and y into —@ and)
—y, the new origin is the centre of the curve.
994. We must now get rid of the term involving the)
product of the co-ordinates wy, by changing the direction of
the axes. For that purpose put (Art. 40),
v=«x cos0—-y sin@, y= x sin@ + y' cos@;
. a(v? sin? 6 +2 2’y' sin @ cos + y” cos’ 6)
+b (a? sin @ cos 0 + x’ y' cos? @ —a’y’ sin? @ — y cosO sin@) |
+¢ (a? cos’@ — 2a’ y' cos@ sin@ + y” sin’ 0) + p (h, k) = 9,
133
or Ay? + Be? + fp (h, k) = 0,
where A = acos?@ —bcos@sin@ + ¢ sin’ @
bss (2),
B = asin’ 0 +b cos@ sin@ + cos? 6
and the coefficient of « y’
= 2asin@ cos @ + 6 (cos? @ — sin? 9) — 2c cos @ sin@ = 0,
which must give a real value for 0, in order that the term
involving ay’ may disappear ;
-“. (© —c) sin20 + bcos2@ = 0,
—b
or tan2@ = :
a-ec
As the tangent of an angle may have any magnitude, it
follows that this equation will always give real values for
20; and if we denote by 2a that value of 20 which lies
between zero and 7, then the positive values of 26 are
2a, W+2a, 2r+ 2a, Si +2a, &e.;
consequently as @ lies between zero and 27 (Art. 41), there
are four values of @, viz.
T 3%
a —+a T a ———e a
3 9 3 + 9 9 + 3
the two former of which determine two. lines at right angles
to one another, and the two latter determine the prolongations
of these lines; so that if we take one of these lines for the
axis of 2’, the other will be the axis of y. Hence there
exists one system of rectangular axes, and one only, proper
to make the product of the co-ordinates a’ y disappear from
the transformed equation.
If however we have, at the same time, 6=0 and
a@=c, tan20 becomes indeterminate, or rather the coeff-
cient of ay’ is identically zero; this proves that we may
in that case take any two rectangular axes whatever, without
introducing the product of the co-ordinates into the trans-
formed equation; and agrees with (Art. 48), for the curve
is then a circle.
134
225. We shall now proceed to the actual determination
of the Axes of the curve. Since
—b
tan 26 = .
Ge
a= CU
/ (a— cy? + roe
*, cos26 =
sin 20 = cos2@.tan20 =
in these expressions the radical may have either the sign
+ or —; because we are at liberty to choose either of the
new axes for the axis of #; but to avoid all ambiguity,
we shall take the radical with a positive sign; then sin20
will have a sign contrary to that of 0.
Hence taking the sum and difference of equations (2), and
substituting the above values of cos26@ and sin 20, we get
(a—c)?+6°
V/ (a—c)? +6?
La ter W/(a+c)?+m},
Lfa+ce—VS(a+e)+mt,
putting m = b° — 4ac.
A — B = (a—c)cos20—bsin26 = = /(a—c)?+8";
226. We have now two cases to consider, according
as m is positive or negative.
First let m be negative, then A and B have the same
sign; and supposing @(h, k) to be of a contrary sign to
A and B, and = —C, the equation is
Ay’ + Bx’ =C, which represents an ellipse
with semi-axes c A WAG and area = ese :
A B V/ 4ac —
If } (h, &) = 0, the equation is satisfied only by w = 0, y = 0,
le. it represents the point which is the origin; and if
p(h, k) be of the same sign as A and B, the equation can
be satisfied by no real values of w and y.
(the term involving a’y’ disappearing
135
Secondly, let m be positive, then 4 and B have contrary
signs; and whatever be the sign of @ (h, &) i.e. whether it
equals + C or — C, the equation will be of one of the forms
Ay’ B# Ba’ Ay
—-——=1, or —-—=1,
e: Cc G C
Pes. boi gy G fe
which represents a hyperbola with semi-axes vit a
If C=0, the equation is y = + BY, which represents
y Ti P
two straight lines through the origin.
227. Next in the equation,
ay’ + bey +ca°+dytexrt+f=0,
let the coefficients be such that 6° —4ac=0, and that the
numerators: in the values of hk and #& are finite, then the
co-ordinates of the centre are infinite, which signifies that the
curve has no centre. In this case, as we cannot by chang-
ing the origin take away the terms involving the simple
|powers of wv and y, our first object must be to destroy the
‘term involving the rectangle wy.
For that purpose put
« cos@ —y' sin@ for x, and w’ sind + y’ cos@ for y,
and the equation becomes
Ay? + Ba? + (dcosO — e sin@) y’ + (dsinO + ecos6) x’ + f = 0,
g, as before, by the con-
‘dition
—6
tan2@ = =
a-c
>)
which gives, since b° = 4ae,
136
taking the radical with the positive sign. Hence by means
of the formule
cosO=V/1(1 + cos 20), sin@ =/4 (1 —cos20), we get
d./a—er/e
d cos 80 — esin@ = —_—_—_____—_ = 9)
a n/a +e .
d/c+er/a
Va+e
Also A=1fa+e+V/(a—c) +b =a+te,
B=h}jat+e-V(a—c)’ +b} =0,
therefore the equation becomes, suppressing the accents,
Ay’ + Dy+Ex+f=0,
D? - =)
4Ak }’
d sin @ 4- ecos@ = = I.
which represents a parabola, latus rectum = —, and co-ordi-
A
D* —-4Af ed d
Wer y= mee 7) » and axis pa-
rallel to the new axis of w In this case the co-ordinates
of the new origin cannot become infinite; for A = a+ ce can-_|
not become zero since a and 6 have the same slon; and if
& =0, then the transformed equation will no longer con-
tain vw; and being solved with respect to y, it will furnish
two constant values for y, so that it will represent two
parallel lines.
nates of its vertex w=
228. Since the general equation of the second order
represents an ellipse, hyperbola, or parabola, according as
6° — 4ac is negative, positive, or zero, it follows that
y a a : we A
pt) +(g-1) * Vg tmeyas
will represent an ellipse, hyperbola, or parabola, according |
as m is negative, positive, or zero; and under this form
137
ot the equation the axis of w is evidently a tangent to the
curve, since when y = 0 each value of a becomes equal to
h; similarly the axis of y is a tangent.
When m=0, the equation becomes
ya Ff & *)
— :& — —2 [= - le=-{)*
( i) Vala ys
: : ‘yt : i
or, taking the upper sign, E + i —i1} =0 representing two
p
straight lines that coincide; with the lower sign we get
(Y v i Any
edi pa ek WL sg A ales SP aad
Vain hk’
or RN Aa
i h
for the equation to the parabola referred to any two of its
tangents as axes. The curve lies wholly between the positive
parts of the axes; as long as w«h and y>k the negative
sign must be taken on the first side.
229. If in Art. 223 the coefficients of the proposed
equation are such that one of the numerators 2ae — bd is
zero, at the same time that 6°=4ac, (which two suppo-
‘sitions make the other numerator 2cd — be also vanish) both
‘the co-ordinates of the centre become indeterminate ; the two
equations (1) in that case are equivalent to a single inde-
‘pendent equation, and the two lines which they represent,
regarding h and & as the co-ordinates, coincide, and there
exists an infinite number of centres all situated in that line.
The proposed equation, with the above relations among its
coefficients, no longer, in fact, represents a curve, but two
parallel straight lines; for, solving it, we get
be+d_ 1
SR NEES EB ee a IR tig tS ac,
ioe b? — 4ac)u* + 2(bd —2ae)4a + @—Aa
ae = VC ) ( )a +p St
138
and this in the supposed case becomes
Se Na tA af
y ae KA d° — 4af,
and therefore represents two parallel straight lines, which
are replaced by a single one if d* = 4af; and become alto-
gether imaginary if d°<4af.
230. We shall now shew how to deduce the nature
and position of curves of the second order immediately
from their general equation, without transformation of co-
ordinates.
The value of y in the preceding Art., since the expres-
sion under the radical sign has either two real factors or none,
may be written (supposing m = b° — 4ac to be different from
zero) either
bea+d 1 $$
th aar aaa = V mw — 8) (a ~ hy); (1)
ie
be+d
_ a
or y ———_—_
2a
—V mi (w= a) + (Cis (2)
be+d
2a
for it bisects all chords parallel to the axis of y. If m be
negative, the value of y in the former equation is real
only from # =g to w=h, supposing h>g, and cannot be.
come infinite between those limits; and in the latter y .
is imaginary for every value of 2; therefore the curve is
limited in all directions, and is an ellipse situated as in fig.
84, where NN’=h—g; and PN’, PN are the first and
last ordinates touching the ellipse at the extremities of the
diameter PP’ whose equation is 2ay+ba~+d=0. If M
be the middle point of NN’, then when # = OM, the irra-
tional part of y, represented by DC, attains its greatest
value; therefore the tangent at D is parallel to PP’; hence
CP, CD are a pair of semi-conjugate diameters, whose mag-
nitudes and inclination being known, the axes of the ellipse
may be determined (Art. 145). Also, w being the angle be-
is evidently a diameter of the curve,
The line y = —
| 139
‘tween the axes, the area of the ellipse = 7.CD.CP sin PCD
=q7snw.CD.MN= (h — 2)?
——————————
qr sin wa/—m
8a
aw sin w
: DEAE jeanne aug
If h=g, then #=g is the only value that makes y real,
and the ellipse is reduced to a point.
| 231. When m is positive, # may have in (1) all values
except those lying between g and A; and in (2) all values
whatever; therefore, in both cases, the curve goes off to
infinity in four directions, and is a Hyperbola.
For equation (1), the curve (fig. 85) is met by the
diameter 2ay+6x+d=0 in the points P, P’, for which
ON’ =g, ON=h; and no part of the curve lies between
the parallels PN, P’N’. Also the equation to the asymptotes
(obtained by developing the irrational part of y and neg-
lecting negative powers of x) is
be+d /m |
x 2a 3 204 ww Set h)y:
both of which lines meet the diameter 7'C in C the middle
point of PP. If h=g, the curve is reduced to two straight
lines coincident with the asymptotes. For equation (2) the
hyperbola is situated as in fig. 86, each branch being convex
towards the diameter 7'’C' which does not meet the curve at
all. When w = a = OM, the irrational part of y receives its
least value represented by CD, and the tangent at D is
consequently parallel to the diameter 7'C. Also the equation
to the asymptotes is
bae+d | /m
2a 2a
=l
(w ey a),
both of which meet the diameter 7'C in C the middle point
of DD’. When 8 =0, the curve is reduced to two straight
lines coincident with the asymptotes.
140
232. If in the general equation c = 0, then m is ae
and the curve is a hyperbola; therefore if a = 0 (in which:
case the preceding results seem to fail) solving the equation’
with respect to #, our conclusions would still hold. Inj
the case however of the square of one of the variables being
wanting, the simpler plan is to solve the equation with re-,
spect to that variable, and we get .
y (be +d)+ce+exr+f=0, |
or, by division,
ar
ba +d’
y=pot+rgt
corresponding to which equation the position of the curve
is that in fig. 87, one of the asymptotes being CP with)
ay equation y = px+q, and the other CD parallel to the axis
ta | of y, whose equation is ba+d=0. If f= 0, the equation’
Wie! is reduced to (pa+q-—y) (ba +d) =0, representing two.
straight lines. |
233. When m=0, the general equation, solved with)
respect to y, becomes
ae beau+d i a
HN | & a 9g ae 3
lar J 2a Sri bar
then if p be positive, « may be taken from zero to infinity,
and y at the same time increases to infinity, both positively
and negatively; but if # be taken negative beyond a certain
limit, y becomes imaginary; therefore the curve has only |
two infinite branches, and is a parabola in the position QPR
represented by fig. 88, Z’P being the diameter meeting the
curve in P. If p be negative, « must be taken negatively
to infinity, and the curve has the reversed position Q’ PR’.
If p= 0, the equation represents two straight lines parallel
to the diameter J'P and at equal distances from it; which
coincide if g = 0, and become imaginary if q be negative.
234, In determining the actual position and magnitude
of the Axes of a conic section from its equation, it will be’
141
always found convenient, as a first step, to transfer the
origin to the centre, when it exists; or to a point in the
curve, when there is no centre. The following are instances
of the principal cases that can occur.
x. I y? —2may + (mM? +n’) 2? -—n’c? =0,
hich represents an ellipse (fig. 84) ;
os Y= ML £ n/c — a,
when =0, y= CD= nce,
when «= CQ=c, y= PQ= me;
° @+RP=CP+CD=c (1+m +n’),
ab= CD x CQ= ne’,
and a® — (a? — 6’) sin’ = cc’, where 4 ACQ = Qo;
which equations give the magnitude and position of the Axes.
UX.) 2s yo —2may + (Mm? — n’*) av’ + nc? =0,
which represents a hyperbola (fig. 85) ;
“Y= MU t n/a —c’,
when v=0, y= CD\/-1=ne\/-1,
when w = CQ=c, y= PR=me;3
- & —P = CP*- CD = c' (1 +m’ —n’),
ab = CD x CQ = ne’,
and a? — (a? + 6°) sin’ = ¢*, where Z ACQ= 9;
which equations give the magnitude and position of the Axes.
: Ex. 3. y? —(m +m’) vy + mmx — c= 0,
pubich ee a hyperbola (fig. 86) ;
Ly= L(m+m’) av &/i( (m' — my) xv* + c.
The diameter CP whose equation is y = 4(m+m’) w = tana.a,
,
falls between the two branches of the curve, and y = ma,
142
y=mu, are the equations to the asymptotes CLS GLa
that if CB be the conjugate axis inclined to the axis 0!
wv at an angle #,
B=4(L'Ca + LCx) = ¥(y/' + ¥y) suppose;
u A 1 VS 14m? J 1+m? 241—mm
sin2BCxe tangBCa m+m :
*, cotanB =
and if the tangent at D, which is parallel to CP, meet the
transverse axis in ¢, we get a and 0 in terms of m, m’, from
b
* = Ct. CD cos BCx = —_—<* , and ~ = cot BCL,
1 + tana tan a
or a? 41 +08 (y' —y)} = b° {1 — cos(¥y’ —y)} = ecos ¥’ cosy.
Cm
Ex.4. y=ma te which represents a hyperbola (fig. 87)
one of whose ipa ai is the axis of y, and the other the line
CP with equation y= ma. Let Cd be the transverse axis,
|
. tan ACa@ = tan $ (90 + PCr) = tanPCa+secPCx —
= Rad a m +m.
Let «# and y’ be co-ordinates of 4A; then
C
PY fy 2 = SSS Sa
/1 +m?
a? = 20(\/1 + m? + mM), :
Cc Rey See ES ER
, , °
ma’ +— = a \/1 +m+tme;
a
and - = cot ACa; .. b? = 2c(\/1 +m” —m).
Ex. 5. y’ —2mye2 + ma? —cxr =0,
which represents a parabola (fig. 88) ;
= M0 + J ca.
The axis of y is a tangent at P the origin, and the line
PY, whose equation is y = mx, a diameter.
Let 4yPV=a, PN=2x, then PV = “and QV =\/ca;
sin a
& AG e
“. Cv =—— x ———; .". latus rectum (4a) = csin’a
sina sin’a
c ‘ : 2a
= ——__—__; and distance of P from the axis = —.
(1 + m*)? m
General Properties of Algebraical Curves.
235. The general equation of the m™ degree between
az and y, ought to contain all the combinations of the powers
of x and y in which the sum of the indices does not ex-
ceed m; therefore when complete and arranged according to
descending powers of y, it will be
ayy” + (by + bya) y?~! + (Gy + Oe + Coa*) y?-* + &e.
-|- (1, + Lav + Lx? -- &c. + L, 2") = 0,
All equations between two variables w and y, which can
be reduced to this form, are called algebraical, all others
are called transcendental; hence arises the distinction of
lines into algebraical and transcendental, according as their
equations are algebraical or transcendental.
236. ‘The classification of lines in different orders, accord-
ing to the degrees of their equations, would be to little pur-
pose, if by changing the axes of the co-ordinates we altered
the degree of the equation. But this is not the case. For,
having given, between w and y, the equation to a line re-
ferred to certain axes, in order to get the equation to the
same line referred to new axes, we must replace w# and y
in the given equation by the values found in Art. 41 ;
and as these values are of the first degree in a’ and y’, it
follows that the degree of the equation cannot be raised by
this substitution. Neither can the degree of the transformed
equation be less than that of the primitive equation; for
if it could, then, by what has been proved, we could not
return from it to the primitive equation, which is absurd.
237. The general equation of any degree comprehends not
only all lines of the order expressed by that degree, but also
144
all lines of inferior orders. ‘Thus the above general equa |
tion of the m™ degree, by making a, = 6,=c¢, =... =1, = 0,-
e . th |
degenerates into the equation of the (2-1). Also the fk
equation of the second degree |
(y— mx —c)(y—-mx—-c)=0
is clearly verified either by putting y=ma+ce, or y=m'a+e,
which represent two lines of the first order; so that the |
proposed equation does not in reality represent a line of the |
second order at all, but two straight lines; or only one |
even of these, if m=m',c=c’. Similarly, the equation of |
the third order |
(y — ma —c) (ay —2’) =0 )
fi
represents a line of the first order, and one of the second |
whose equation is ay —a* = 0. And, in general, according as |
a proposed equation of any degree is not, or is capable of |
being resolved into factors which are rational with respect to |
the variables w and y, it wiil represent a single line of the }
corresponding order, or several distinct lines of inferior orders. |
oD SO straight line cannot meet a curve of the n™ order +
in more than 7 points.
Let the co-ordinates be transformed so that the pros |
posed line may be the axis of w, and let V=0 be the re~ |
sulting equation to the curve; in order to determine the
points in which it is intersected by the straight line, we |
must put y=0 in the equation V=o0, and the corresponding |
values of w will be the abscisse of the required points. |
But V=0 being of the m™ degree, the equation for deter. |
mining «x will be at the most of the n™ degree; therefore
# cannot have more than 2 values, and there cannot be more
than m points of intersection 5 but there may be fewer than
nm, for the equation for determining & may be of a degree
inferior to n, and may have equal or imaginary roots.
239. The general equation of the n™ degree between two
variables, when complete, contains 1 + 2+3 + &c. + (2 +1),
or 4(u +1) (v +2), arbitrary constants, in which, since we
145
may divide the whole equation by one of them, there is one
superfluous which might be suppressed ; consequently the
number of independent constants is
$(m+1)(m+2)-1, or tn(n + 8).
Hence a curve of the » order may be made to fulfil
-3n(m +3) conditions ; as, for instance, to pass through
$n(n + 3) points ; for, giving to # and y their values at each
of the given points, we get dn(m + 3) different equations
by means of which the values of the constants may be de-
termined. Hence a curve of the second order may be deter-
‘mined so as to pass through five given points; as will be
‘seen in the following Problem
| 240. To determine the conic section which
‘through five given points.
|
|
|
shall pass
| Take the axes of the co-ordinates so that each axis con-
tains two of the given points; and let y,, y,, be the ordinates
of the points situated in the axis of Y3 ®, &, the abscissee
of the points situated in the axis of w; and ay, Y35 the co-
ordinates of the fifth given point. Then substituting suc-
sessively the co-ordinates of each of these points in the place
of w and y in the general equation (where every coefficient
s divided by the constant term),
ay’ +bey+ca°+dy+en+1=0,
ve get the five equations
0, ays +dy+1=0,
CH, +e%,+1=0, Cx, + et, +1=0,
hich give for the five unknown quantities, the values
Yi + Yo it V, + B
Sie 10 =) — -5 C= > €=—-— 73
{iY Y1Y2 vB, UX,
1 Uo(Y.— — ve. Vs —-X— Xv
ba — ts Y2 oe, RO}
V3 Y3 Y1Y2 WV) Ve
10
146
Now provided no three of the given points be in a straight |
line, none of the quantities 4%, @, &c. is zero; therefore the
above values of a, 6, &c. are neither infinite, nor indeterminate,|
and none of them has more than one value; therefore through}
five points, provided no three be in a straight line, a conic)
section, and only one, can be made to pass.
i
[
f
,
‘
|
¢
241. Every curve of the n order which passes through!
n(n +8) —1 given fixed points, will also pass through,
n(n — $) +1 additional fixed points.
Since the given points are one fewer in number than
what would be sufficient to completely determine a curve!
of the n™ order, an infinite number of such curves may be
described through them. Of these, let us consider any twe|
whose equations are M = 0, M’=0; then the equation]
M’ + »M = 0, (1) (where yw is an indeterminate constant) will
include all the curves of the »™ order that can pass throug!
the given points, since the equation of every such curve
could involve only one undetermined constant. But equa
tion (1) will be satisfied by every pair of values of w and 4
which satisfy M=0, M’=0; therefore the curve (1) wil}
pass through the m? points of intersection of AZ = 0, M’ =0
that is, all the curves will pass through the points of intersee
tion of any two of them; therefore all the points of inter
ATS) Bee oival
section must be fixed points; and the lis
points will determine the remaining points of intersection|
Hence every curve of the m™ order, besides passing througl
a number of fixed points one less than the number sufficient ti
completely determine it, will also pass through an additiona
number of fixed points such that added to the former it make
up 7”, the entire number of points in which two curves 0
the n™ order can intersect one another. |
Hence 8 given points of a curve of the third order wil
determine a ninth point of the same curve; and 13 give
points of a curve of the 4th order will determine 3 ne)
points of the same curve.
147
242. To find the position of the centre of any curve.
The centre of a curve is a point C (fig. 24), such that
any chord of the curve PP’ drawn through it, is bisected in
it. (It must be observed, however, that if PP’ mect the
curve In more points than two, it is sufficient that these
points combined in a certain order should be two and two
equally distant from C.) If the curve be referred to any
two axes originating in C, and PN, P'N’ be the ordinates
parallel to Cy of the extremities of a chord, we see from the
equal triangles PCN, P’CN’, that these ordinates are equal
and of contrary signs; the same thing is true for the abscisse
lof P and FP’; as well as for the extremities of every other
chord passing through C. If therefore p(w, y) =0 be the
equation to the curve, and if it be satisfied by w=a, y=),
it must also be satisfied by w= —a, y=-—b; that is, it
must be such as not to alter when the signs of the two vari-
‘ables are changed ; and conversely, if it have this property,
the origin is the centre of the curve. When f(a, y) = 0 is
algebraic, it cannot have the above property unless the di-
mension of every term be even in an equation of an even
degree, and odd in an equation of an odd degree; for in
the former case the equation is not at all altered by replacing
ve and y by —w# and —y; and in the latter (in which case
che equation cannot have a constant term) the sign of every
‘erm will be altered, and therefore the whole equation un-
iltered. Hence to find whether a proposed curve admits
of a centre, we must refer it to parallel axes through a new
origin having co-ordinates h, k, by putting w=a' +h, y=y' +k;
ond equate to zero the coefficients of all the terms which
‘te of a dimension different (as far as regards odd and even}
rom the degree of the equation; if these conditions can all
ve satisfied by real and finite values of h and k, the curve
as a centre, and h and & are its co-ordinates; in the contrary
ase the curve has no centre. Of this process we have an
Kample at Art. 223.
| 243. The locus of the middle points of a system of
arallel chords of any curve, is called its diametral curve.
10—2
148
If the curve be of the »™ order, the points of intersection,
with its ordinates real or imaginary will be in number 73
and their combinations on the same indefinite line will form|
n(n —1) different chords, and as many middle points, and,
Tees the diametral curve, since it may be met by an|
indefinite line in 4n(m—1) points, will have an equation of;
the degree 4n(m —1). For curves of the second order, since)
m = 2, the ‘diametral curves can only be straight lines ; for
curves of the third order, the diametral curves are also of
the third order.
;
244, To find the locus of the middle points of a system
of parallel chords of any curve.
Let the chords be parallel to a line through the origin
whose equation is y= ma, and let (a, y) =0, be the equa-|
tion to the curve; also let a’, y’, be the co-ordinates of the
middle point of any one of these chords, and take it for the
origin without altering the direction of the axes, and therefore
put « +a for xv, and y'+y for y; then the transformed
equation to the curve is @(w’ + a, y' + y) =0, and the equa.
tion to the chord is y= ma. Hence the values of a, cor
responding to the points of intersection of the curve and
chord, result from the equation @(e’ + x, y +m) = 0, on
a" + pa") + pra"? + &e. + P, = 0, suppose ;
and because the origin bisects the chord, this equation mus
be satisfied by — a,
a — pa"! + pow"? — &e. + (— 1)"p, = 03
between which two equations if we eliminate 2, we obtait
a relation between w and y’, which is the equation to th
required locus.
245. Thus if d(#, y)=0 be the general equation o
the second order,
ay’ + bry +ca°+dy+texrt f=0,
149
putting w= a! + x, and y=y + ma, we get
a(y + ma) +b(e + 2’) (y+ mx) +e(x +a’)?
+d(y'+ma)+e(a+o')+f=0,
-and because the values of w are to be equa] and of contrary
signs, the term involving the first power of w must disappear ;
". 2amy' + b(a'm + y') + 2ca' + dm+e=0,
or y (2am +b) +a’ (2c + bm) +dm+e=o0,
the equation to a straight line. Hence there will be an
infinite number of diameters corresponding to the various
values of m.
If the diameter is to be perpendicular to its chords, we
must have
2c + bm > 2(e -a)
1+ m | -————_} =0, or m shart appr hee lO
which will necessarily give two real values of m; hence there
are generally two diameters which bisect their ordinates per-
pendicularly.
If (x, y) =0 be the general equation to curves of the
third order, and
p(w +a, y + ma) = x + pia + px + ps = 0,
then also a — pa’® + p.w —p, =0;
therefore, adding and subtracting, a + p, = 0, piv? + p,=0;
-". ~3= P,P. 1s the equation to the diametral curve.
246. Not only are Algebraical curves distributed into
ders according to the degree of their equations; but also the
ifferent families of lines are investigated, which may be com-
ised amongst those of the same order; and even the dif-
erent species of each family, if necessary. The individual
nes of the same family, or species, are then classified ac-
ording to certain characteristics easy to be recognized, which
ompletely distinguish them from one another; and lastly it is
150
endeavoured to determine the form and properties of each of —
them. This has been here effected for equations of the first
and second degrees; the former gives only straight lines, as
has been said; the latter gives three species of curves suf-_
ficiently distinct ; viz. the parabola, which has no centre and
the ellipse and hyperbola, both of which have a centre, but
only the latter has asymptotes.
The enumeration of lines of the third order was first”
made by Newton, who found 72 species comprized in 14
divisions; Stirling added 4 species which had been omitted ;_
and, lastly, Cramer added two more, making in all 78 species. _
On tracing Curves from their Equations.
247. When a curve passes through the origin, the angle
at which it cuts the axis of « may be determined by taking
the limit of ~ when « = 0, which will be the value of the
x
tangent of that angle.
Let AP be a curve passing through the origin A (fig. 72),
P a point in it near A with co-ordinates AN = a2, NP =y;
1
draw the secant AP, then tan PAN = “a now let P move
up to and coincide with 4, then the secant AP coincides
with AZ’ the tangent to the curve at 4, and
tan TAN = limit of tan PAN = limit of ! , when # = 0.
Lv
Hence the angles at which a proposed curve cuts the co-
ordinate axes may always be determined ; for we have only
to transfer the origin to one of the points in question, and
in the transformed equation take the limit of re! by putting
wv |
U
v= 0. '
248. In tracing curves from their equations, whenever
y is given, or can be found, in an explicit function of @,
it will be best to use algebraical processes alone. |
151
First, determine the points where the curve cuts the
axis of x, and its shape at those points. For that purpose
transfer the origin of co-ordinates, if necessary, to one of
the points ; expand y in a series ascending by powers of 2,
and let
y = av” + ba"* + &e.
ans (/ :
If m < 1, limit of st co; therefore the curve is per-
wv
pendicular to the axis of #, and immediately afterwards is
concave towards that axis.
eae q
If m= 1, limit of ue a; therefore the curve cuts the
a
axis of w at an angle whose tangent is a, and immediately
afterwards is situated above or below the tangent, i.e. is
convex or concave towards the axis of w, according as 6 is
positive or negative.
ants q
If m>1, limit of ite 0; therefore the curve touches the
v
axis of aw, and immediately afterwards is convex towards
that axis.
Similarly, the form of the curve at all its other inter-
sections with both the axes may be found.
ww
and to that end expand y in a descending series of powers
of w (on the supposition that both w and y are very great),
and let
Secondly, determine the nature of the infinite branches;
td
Y = au" + Oe + 0... FOU +f +> +o... 5
v
“ y= an" + bu +...... +ex +f is the equation to the
asymptotic curve, above or below which the given curve is
situated, according as g is positive or negative,
If m= 1, the equation to the asymptote is y=ew +f,
representing a straight line; and the curve is situated above
152
or below the asymptote, according as g is positive or negative ;
consequently, as the curve will be convex towards the recti-
linear asymptote with which it continually tends to coincide,
it will ultimately be convex or concave towards the axis of @,
according as the first of the neglected terms is positive or
negative.
In this case the infinite branch represented by
y=eutfrt é Se re
is said to be hyperbolic.
If m> 1, the infinite branch is parabolic; and it is con-
cave or convex towards the axis of a, according as its
asymptote 1s concave or convex towards .that axis.
249. Having thus found the figure of the curve at the
points where it cuts the axes, and also when wv and y are
very great, the intermediate parts may generally be traced.
For the actual position of the maximum or minimum ordi-
nates and points of contrary flexure, recourse must be had
to the methods of the Differential Calculus.
If the equation to a curve can be resolved, and give
for y the values V, V’, &c. functions of # and constants,
we must trace separately each of the curves represented by
y=V,y=V’, &e., all of which will be particular branches
of the proposed curve. ‘The branches cannot terminate ab-_
ruptly; they will either go on to infinity, or the ordinates
will become imaginary, in which case two branches will be
united and mutually continue one another.
Ex. 1. ‘To trace the curve whose equation is ay’ = (a?—a’)?
Since the equation does not alter when —~ is written
for wv, the curve is symmetrical with respect to the axis
of y; also for any value of w, y has only one possible
value, and is always positive. When # =0, y =a, and as
153
w@ increases either way, y diminishes; therefore a is a maxi-
mum value of y, and the curve cuts the axis of y at Bat
right angles, and is concave to the axis of wz. When v= a,
y=0, and the curve cuts the axis of a at right angles
at A, because if we remove the origin to that point by
making w=a+a', we get ay =(2aa' +a)%, and theref
Ewezratsz, g 2 AG +2), a 1erefore
3 A Me.
o y ro, (2a+2
limit of =, = limit of doar OT When w#>a, the
v ax
equation becomes ay’ = (v* — a*)*, and as wv increases, y in-
creases, till w is very large, when the relation between
them approximates to
ary = Bil— 7 3j;]—— me = 73 — — —3
a 83
. ay = a3 is the equation to the asymptotic parabola ZOZ,
below which the curve lies, because the second term of the
expansion of y is negative. Hence the figure of the curve
is that annexed, having a point of inflexion at P; for the
curve is concave to the axis of w at A, and afterwards con-
vex because the parabola with which it tends to coincide is
so. ‘here is of course another point of inflexion at P’
i
>
Ex. 2. To trace the curve whose equation is
e—38
= fig. 74).
Lrg Dip (ai aye
When #=0, y= — ~, and the curve cuts the axis of y
~
at D; as long as w<1, y is negative and becomes infinite
when w = 1; therefore the ordinate BE corresponding to x = 1
is an asymptote, and we thus get the branch DE.
When « is between 1 and 2, y is positive, and becomes
very great both when w is a little less than 2 and a little
greater than 1, and gives the portion EGF’,
When #=2, y is infinite, so that the ordinate FF" is
an asymptote.
154
When « lies between 2 and 3, y is negative and gives
the portion #7.
When #=3, y =0, and the curve cuts the axis at Hl. |
When x > 3, y is positive and gives the portion H&A; |
and when & is very great,
v 1 0 ;
y=; — =< a 2 /
= LB — 32 “2—s A
therefore the axis of w is an asymptote.
When w is negative, the equation is
eo+s i |
J = (v +1) (@ +2) |
therefore y is always negative, and diminishes as # increases,
and becomes 0 when a =o and gives the branch DL.
Ex. 3. To trace the curve whose equation is
ay = x +ar/2au— x* (fig. 75).
Taking the radical, which of course admits of a double
sign, first with a positive sign, we have when «= 0, y= 0,
me as q
and limit of uae 0; therefore the curve passes through the
v |
origin A and touches the axis of #; when w=a, y= 2a,
and when «=2a, y=4a, beyond which y is impossible;
we thus get the portion of the curve AED, the ordinate.
CD being a tangent at D. Again taking the radical with
a negative sign, so that
ay = @ — vr1/2a0 — 2”,
e e e q
”7=0 gives y=0, and limit of —~=0, as before; as long
v
as © Y Wak y)3 ( )
herefore the proposed equation becomes
f ae’? —
we eae , , , = 3 av’ 19
= nae 4 (e@ y) +(e +y)°t Wit +3a y*),
vhich can evidently be solved with respect to y’; and it shews
hat the new axis of wv’ is an Axis of the curve.
By the same transformation the equation Saxy =a +43
3a :
becomes SE (a? —y") = x3 4 30 y’?;
~
nd the new axis of 2 is an Axis of the curve:
11
162
12. To express by polar co-ordinates the equation
y? = 4a(a +2) + tan’a (20+ 2)".
Adding «” to both sides we get
or +y = (2a +2)’ (1 + tan’a) ;
“. © cosa = (2a +7 cos 0)’,
or £rcosa =2a+4+1rcos@.
13. The locus of the middle points of all chords of a
circle that subtend a right angle at a fixed point in its plane,
is another circle (fig. 01).
Take O the fixed point for the origin and two lines O#,
OF, at right angles to one another and cutting the circle,
for the axes; and let its equation be
(a — ay + (y—b) = 673
then OF =y' =b + \/c?- a’,
OF =a =at/ce—b’
Let G be the middle point of any chord EF, and D the
middle point of OC, C being the centre of the circle; then
DGt=1(y -b) +4 (@' -a)? =1@ce-a-B);
therefore the locus of G is a circle, whose centre is the middle
point of OC.
14. Three given circles being traced upon a plane, te
shew that any three angles, each of which contains two o
the circles, will have their vertices in a straight line.
Let A, B, C, be the centres of the three circles (fig. 92)
a, b, c, their radii; and let lines touching the two circle:
about A and B meet in F, and those touching the circle
about B and C meet in D; join FD and produce AC t
meet it in HL. Then is £F the intersection of two line
touching the circles about 4 and C3; for
Ak AE BG FA BD
Al A Bh BD a b a
CE. BE CE. FBICGD bc ia
163
15. To find the co-ordinates of the points of intersection
of two circles which cut one another.
Let their equations be
ie oy yp = ¢, (@ vf a)? at y? fab ¢”,
‘so that the axis of x joins the centres; then it may be easily
proved that the co-ordinates of their points of intersection are
: a? + 2 ~ ¢”
& ee
2a ?
eee ae a Me SS Declan AO
Meo, 4 @+ctc),(@1c—c). (are —c).(c+e'—-a);
tom which all the common theorems in geometry, relative
io the intersections and contacts of circles with one another,
may be deduced.
16. In the sides 4¥ =a, AP = b, (fig. 14) of a given
riangle AP_X, take two points M, N, such that
XM AN _
M4 wp ™
nd join MN; then all the circles described about AMN,
or different values of », will have a common chord.
It will be found that the equation to the circle referred
9 AX, AP, as axes, putting 2. XAP =, is
(w* + 2ay cos w + y’) (n +1) —ay-nbe=0;
id in order that this may be satisfied by values of
wand y
dependent of , we must have
w + 2ey cosw+y? = ay = be.
These equations give real values of w and y, and con«
quently determine a point in the circle whatev
erefore all the circles have a common chord
mis ay = be.
er 2 be;
; and its equa-
11—e
164
17. If in a regular polygon of n sides, lines be drawn
from the extremities of a side to those of any other side so
as to cross each other, the locus of their intersection is a circle, |
29
whose radius = sa cosec —.
. n
Let AB, BE, CD, be three sides of the polygon; pro-
duce AB to F, and let AD, BC, intersect in P; then since
9
arc AB = arc CD (fig. 93),
, PAB = CBE: and 2 EBF = =;
29 : :
2 APR =-—, and locus of P is a circle;
n
: : Atr : |
, and AB =a is a chord subtending an 4—., at its centre;
n
-, its radius = La cosec —.
18. To find the locus of a point from which, if thre
perpendiculars be dropped on three given straight lines, the
points of intersection shall always lie in a straight line.
Let the three given straight lines intersect one anothe
in the points A, B, C (fig. 94), AC = b, AB=c, LBAC=a
x, y, co-ordinates of P referred to AC, AB, as axes. PM
PN perpendiculars on the axes; join MN cutting BC in Q
and join PQ, then PQ is to be perpendicular to BC. =
equations to BC and MN are respectively
Y= -"X4e= —nX +c (1) suppose,
2
+ & COS Ww
Views of us taae aes X+Y + €COSW3
@ + Yy COSw
therefore the co-ordinates of their intersection Q are
i (y—¢+ 7008) (@ + Y COS w) y’
apy ae em anty 7 6) 9") Ce
(y + @cosw) —n (vt + ¥ Cos w)’ Y
165
The equation to PQ is Y—y = a (XY — x); and as
ih v
this is to be perpendicular to (1),
rag)
-- 1—mcosw + (cos w — 7)
0 (Art. 338).
7
=n
Therefore, substituting for Y’ and X’ their values, and
reducing, we get
x+y’ + 2xy cosw=bx + cy,
the equation to the circle circumscribed about the triangle
ABC.
The converse of this admits of an easy geometrical proof.
Drop the three perpendiculars PM, PN, PQ, upon the three
mes, and join MQ, QN. Then z MPN = supplement of
MAN = BPC,
*. BPM =CPN; consequently BQM = CQN,
ind therefore MQ and QW are in the same straight line.
19. ‘To find the diameter of the circle represented by the
‘quation
w+ 2xeycosw+y =ba+cy.
In order that this equation may represent a circle, the
ixes must be inclined at an angle =w; also the origin is
. point in the circumference; and making w=0, we get
'= AB = c, (fig. 94); and if y=0, # = AC =); hence dia-
BC r/c +b? — 2be COS w
sin w SIN w
neter of circle =
20. If two parallel planes revolve in their own planes
n the same direction about fixed points 4, B, with equal
elocities; the curve traced on the first by a pencil fixed
erpendicularly in the second at a given point P, will be a
ircle.
Let A be the origin (fig. 95) d4B=c, BP=a, AP = 1,
vAP = @; AB’ being the position at any time of that fixed
166
line of the first plane which coincided with AB, at the instant
that P was also in AB. ‘Then since PB is parallel to AB’,
— cos@ = cos APB = ‘pioranpa weiss
2ar
or 7° + 2ar cos + a? = c’, the equation to a circle.
If the planes revolve in contrary directions, it may be
similarly shewn that the locus of P has for its equation
1 ‘
x? — 2arcos@ + a’? = — (a? —7")’.
es
21. To find the equation to the curve traced by a nail
projecting out of a vertical wall on a circular board that rolls
on level ground at the foot of the wall with its plane vertical,
Let N (fig. 96) be the nail, Aw, Ay, the co-ordinate axes
fixed in the board, and suppose the axis of y at the commence-
ment of the motion to be vertical and to pass through NV; the
board is supposed to have rolled through an 2 eAQ=¢@, where
AQ is horizontal; wAN=0, AN=r, NB=a, AQ=6;
then 4B =cg = horizontal space passed over by the centre ;
P=a'+eg?; but ZNAM = NAB- «AQ;
therefore the required equation is
|
a H : |
= sin7* - ~ — a”.
:
22. If x, 2’, be the radii of two circles in one plane aul
a the distance of their centres, to find the locus of the points
from which the two circles appear equally large.
Let a common tangent to the two circles intersect the line
joining their centres at a distance d from the centre of the
smaller circle whose radius is 7, then d = ar — (7 — 7). Let
v, y, be co-ordinates of any point of the locus, referred to the
above point of intersection as origin, # being parallel and y
perpendicular to the line joining the two centres; then
(7 -d)?+y’ ph
fy ?
(dta—a)+y 7”
167
_ (a? fh. y’) (9 a7 ¥*) —~ On Sx? -_/" (d + a) = 0,
,
nov, arr
or PY Has oe y ea & oF 1. ° 9
— 7"
the equation to a circle.
ti
3. Having given the lengths of two tangents to a para-
bola at right angles to one another, to find its latus rectum.
Let QP = b, QP’ =c, (fig. 31)
2a
then —~ = 1+ cos@ = 1—cos@;
SP ,
2a
SP’
Vide
therefore by Arts. 83 and 86, a. sq 1
i ’ SQ? . P’ P? 4b?”
A Se
PP (b? + c)2
:
24. ‘To find the locus of the intersection of two normals
to a parabola at right angles to one another.
If m be the tangent of the inclination of the normal to
the axis of a, its equation is (Art. 80)
y+ 2am = m(x — am’);
J e I e
and changing m into ——, the equation to a second normal
: m
perpendicular to it is
Hence, adding and subtracting, we get
(Z-m)
y=at\t——m “—-8a=al|——-m);
J m ‘ m s
. y =a(v — 3a),
the equation to the required locus, which is an equal parabola.
168
25. To draw normals to a parabola through a given
point.
Let h, k, be the co-ordinates of the given point, a, y
those of a point in the curve through eh one of the |
normals passes; then
k-y=-= (h-a), y = 402;
and eliminating 2, we get y* — 4a (h — 2a) y — 8a°k=0, which
compared with y? + qy +7 = 0, gives
2 3 4
Agee de U 16a {i Lae «Nie aay
4 27 27a
~
4,
hence (‘Theory of Equations, Art. 90), when k* = —— (h-2a)’,
27a
two normals may be drawn through (h, &), for then y has
only two real different values; and according as k’ > or
4. e e
oa ate: (h — 2a)’, y has only one, or three real distinct values;
a
~
and one or three normals can be drawn through the given
point. This is the same thing as saying that through a given
point one, two, or three normals may be drawn according as
the point lies without, upon, or within the evolute.
26. To find the radius of the circle which passes through
the intersections of the tangents at the extremities of three
given focal distances of a parabola.
The diameter of the circle (since it is described about the
MS SM. SL
triangle MLS) = SAUL = RFE)
_ VS SP.SP’.\/SP.SQ VES om SP
/ SP.SA
27. If one side of a triangle and two others produced be
tangents to a parabola, and the points of contact be joined, a
|
169
triangle will be formed whose area is double of that of the
| exterior triangle.
Segment PQR = 2A PTR (fig. 97) (Art. 103),
or APQR + 3(APUQ + AQVR) =2APTR;
. APQR = 3(ATUV + APQR),
or APQR =2ATUVD.
28. If in any segment of a parabola a polygon be in-
scribed having the same base as the segment, the sum of the
| cube roots ae the areas of all the partial segments standing
| upon the sides of the polygon, is equal to the cube root of the
area of the whole segment.
| Let y, y, y", be the ordinates of three points P, Q, R,
(fig. 98); take s the middle point of the chord PQ, and draw
msn parallel, and Qn, Mm, Ns perpendicular to the axis Aa;
2
AM MEY.
16a
then sN=1(y+y’), «.
y+y?
8a
and AN =
?
(y'-y)
~ MN =ms =
l6a
also Qn =1(y' -y),
ie NS
.. segment PQ = ems. Qn = (y - 9)".
244
similarly,
P oe I\3 ny x 3
segment QI = eae segment PR = as 3
Ps V/ sez. PQ + V/ seg. QR = a/ seg. PR. |
29. ‘Two tangents a, b, to a parabola intersect at an
angle = w, and a circle is described between the tangents and
the curve; to find its diameter.
170
The equations to the parabola, and circle, referred to the
tangents as axes, are
v y ste
therefore for the points of intersection of the curves,
ar? inane _ a
e+o(1—a/2) ~2sin}wv/ab(1-A/2) =r cot hus
a a
and in order that the circle may touch the parabola, the two
values of x must be equal in this equation,
“(a+b 4+2V/absin he) (b- 7 cothw) a
= ab(b+2r/absinkw + asin®?Lw),
; : ab sin w
which gives 27 = —————__________,
ah
a +b+2/absinlw
30. If one side of a triangle and the other two pro-
duced be tangents to a parabola, and each of the angles of
the triangle be joined with the point in which the parabola
touches the opposite side; the three straight lines thus drawn
will intersect one another in a point the locus of which, for
different parabolas, is an ellipse circumscribing the triangle.
ABC the given triangle (fig. 99), RPS a parabola touch-:
ing the base in P and the two sides produced in R, S;
AR =r, AS =, and a, b, c the sides of the triangle. Then
taking AR, AS‘ as axes, the equations to the parabola and to
its tangent:at P (h, k), are
V
a/2 /y y
— t+ —i—) aarr ¥ eet LI
ip N= ; rh «/sk
f ss Ara b e
“1 €=4/sk, b=/rh: and Ft gM UB):
Now the equations to BR, C\S, are respectively
v
Sah ate Piao J
b)
? Cc §
171
therefore for their point of intersection O we get
y Hy Ws x
Ss ay oe, My
Us) tora A as Op C aan Exeale
oe a | eerie een Pn — Cae ge oe : Oa Ie bec MTR Gat ret
w@ \GeAS Ovary wv er ‘sb mos GL Tk
which shews that the line joining 4 and O passes through P
Also eliminating + and s between (1) and (3),
; Dest
8 e
we get by (1 _ | +O (1 ~ *) = UY,
the equation to the locus of O, which represents an ellipse
passing through the points 4, B, C.
31. To describe a parabola which shall touch three
straight lines, one of them in a given point.
Let R be the given point in AC (fig..99), and AB, BC,
the two other given lines touched by the parabola in the
points S and P.
Then using the notation of the preceding problem, we
shall obtain the equation to the parabola referred to PB and
Px which is parallel to AR, as axes, by putting
Par \ ewenas | B91. 0?
WI VOCS Yay Y= Yi 5
a ” a s
therefore the equation is, suppressing the accents,
IM, RE we by 0b 2)?
OU ee ee eT 5
Gees (eS Ne os
; a 2a ~~
or, reducing, y = Fie ot Bis V/(r - b) x.
Hence, proceeding as in Ex. 5, Art. 234, if PV be the
. e ° a
diameter whose equation is y=- a, and 2 BPV =a, the latus
r
rectum
4.07 sin’ a@ 4a°r (r — 6) sin? C
= (Tr —_ b) 5 = 2 2 3 (1).
7 sinC (7° —2arcosC + a’) |
(r - b) (8-6)
(rb +sce-a’)i-
or L = 4sin’ 4 J bers
32. ‘To describe a parabola touching three given straight
lines so that its latus rectum may be the greatest possible.
Since the latus rectum vanishes when r = 6, and also when
=C, it must admit of a maximum as the point P moves
from C' to B (for the focus describes a circle passing through
A, B, C), and the corresponding value of 7 is given by the
equation
+ 1° (acos C — 2b) + ar (bcos C — 2a) + ba® =0;
which will have three real roots, as each side will be touched
by a parabola whosé latus rectum is a maximum.
Hence if the magnitude of the latus rectum be given, each
side will be touched by two parabolas, having latera recta
of that magnitude; and we see that equation (1) for finding
r would be of the sixth degree.
If we suppose the two lines 4B, BC to become co-
incident, then JS’ and P coincide with B, and -the parabola
touches BC in a given point for which BC =a,
4a°7? sin? C
and latus rectum = —>——_———; -
(7? — 2ar cos C + a*)8
33, ‘The directrix of every parabola that touches three
given straight lines passes through the intersection of the per-
pendiculars dropped from the intersection of every two lines
upon the remaining one (fig. 94).
P a point in the circumference of a circle circumscribing
triangle ABC, and therefore the focus of a parabola touching
the sides 4B, AC, BC; «, y, its co-ordinates referred to AC,
AB as axes; PM, PN perpendicular to AB, AC; then the
equation to MN, which is a tangent at vertex of parabola,
Is
+zcosa
Y=mxX +c, when m = A ike etl
v + ycosa
.. equation to a line through P parallel to MN is Y-y=
m (X — 2).
Let M'N’ be the directrix, and therefore parallel to MN
and at the same distance from it as F is;
, y 1
. AN =2(#@+ycosa)-x+—=a@+4+|2cosat+—} y,
m m
AM =2(y+xcosa) -—y+me=y + (2cosa+™m) 2,
or, substituting for m its value,
(a? + y? + 2xy cosa) cosa
AN’ = :
y+ xcosa
(a? + y’? + 2avy cos a) cosa
AM = 21 _;
v+ycosa
. AX (y + vcosa) + Y(# + ycosa)
= (bx + cy) cosa is the equation to NM’,
(.. the equation to the circle is 2 + y?+2xvycosa=bxe+cy)
which equation is satisfied, whatever be # and y, by
X + Y cosa = ccosa,
X cosa + Y = bcosa;
Me ae sin? a
Y sin’ a
I
cosa(c — bcosa é ; ,
( ) which determine the intersec-
cos a (b — ccosa)
tion of the perpendiculars.
34, If the vertex and nearer focus of an ellipse be fixed,
whilst the centre assumes all positions in the indefinite line
passing through them, the curve will successively become a
parabola, circle, limited straight line, hyperbola, unlimited
straight line, hyperbola, and _ parabola.
174
the equation is
and it assumes the following forms for different values of c;
= ©, y°=4p~a the limiting parabola,
c= 0, a circle.
When ec is negative, the equation is
—2c a
ya Pp ( P
p-e
c4p p, a hyperbola exterior to the limiting parabola; and for
c=, the curve is again the limiting parabola.
35. ‘To find the locus of one end of a given straight line,
whose other end, and a given point in it, move in straight
lines at right angles to one another.
AP the given line =a, B the given point in it,
PB=b, CN=a2, NP=y, LPBN=6; (fig. 101) then
ev=acos0, y=bsin8, ie!
ipa aed , ‘
— + ey 1, the equation to an ellipse.
ERT
If the rectangle CO be completed, and PO joined, PO is
a normal to the ellipse at P; for
GN GN BN bb ®&
CN “NG bua iaineds
Calling AS'=p, SC =, and taking the vertex for origin,
175
Hence if a line, whose length is the semi-major axis of
an ellipse, have one end in the curve, and the other in the
minor axis; then (1) the part cut off by the major axis will
_always equal the semi-axis minor, and (2) the locus of the
intersection of the perpendicular to the minor axis through
one end, with the normal through the other, will be a circle.
36. ‘Two given circles touch each other internally; to
. shew that the locus of the centre of the circle which touches
each of them, is an ellipse having their centres for its foci.
SJ and H the centres of the circles, P the centre of a circle
touching both; join SP passing through the point of contact
B, and HP passing through the point of contact 4 (fig. 102);
thn SP + HP=SB+ BP +(HA- AP)=SB + HAA,
which is constant ; therefore the locus of P is an ellipse.
| 87. Ifa, B, y be the angles which the transverse axis,
and the focal and central distances of any point of a curve of
| the second order, make with the tangent at that point,
tana.tany = tan’ 3.
Let SPY =6; CPY =v, STP =a, (fig. 41)
SG cos p-
$ na hed 3
SP cosa
then e =
NG :
and —— = tana.tan(y —a)=1-—e*=1-———_ ;
therefore, reducing, we get tan a tan y = tan? 2.
334 Lhe products of the alternate segments of the sides
of a polygon described about an ellipse are equal.
Let p, g, 7, s be the lengths of the semi-diameters re-
spectively parallel to the four tangents at P, Q, R, S, (fig. 103)
the proof being the same whatever the number of sides ;
O,Q iy q? O.P Rae O.R gp? CO See 3?
” O.P.O0.R. OS .0,Q = 0,Q:.0;,P .0O:R . OS:
(Art. 155);
then -
176
|
In the case of a triangle ABC whose sides touch the ellipse.
in the points a, 3, y, we should get Ap. Ca.By=Ay.Ba.CB, |
which shews that the three lines j joining the points of contact |
with the opposite angles, intersect in a point. |
39. If @ be the acute angle between the tangent and.
focal distance at any point of an ellipse, the distance of that.
point from the centre is \/a? — b? cot?@ (fig. 42).
2 PE? |
For pad (51) <0 — 2 (, ~1) =0°—B cot? 8, |
P |
CQ’
40. To find the locus of the intersection of the normal to |
'
an ellipse and the perpendicular upon it from the centre. The |
equation to the normal at any point is
:
(y — ma) a" + mb + (a — B) m= 0,
and the equation to a perpendicular upon it from the centre is’
1 } : a
y =——2#, which gives m= —-; |
a y
therefore, substituting, the equation to the required locus is
(a? + y?) Jay? + Bx = (a? — 6) wy.
41. A given triangle has always two of its angular points :
in two given straight lines; to find the locus of the remaining ©
angular point.
Take the given lines for the axes, and let
ZAOB=w0, CM=x, CN=y; 4 OAB = @, (fig. 104) ;
“. ysinw = bsin(A + 9),
wsinw =asin(B+nr—d-w)= —asin(4+$+C +o),
.. bxsin w= —a cos (C+w).ysin w— ab cos (4+ ¢) sin(C+w), |
“. sin’w iba + acos (C+w) y}*= a’sin? (C+) (0 — y’sin® w),
the equation to an ellipse of which OQ is the centre. |
42. "To find the Pisa of the middle point of a chord of
constant length in an ellipse.
177
Let QV=c, CV=r, ACV=8, (fig. 45); then
QV? CV?
CD?” CP
b?
1 ~e* cos?@ ”
= 1
But, CP? =
CD = a2 4B? CP? a 1 E+ B) & c0°9
1 — e* cos’?
9
c* (1 — e* cos? 6) ri r” (1 — e* cos? @)
e , =
a” — (a* +b”) e* cos?@ b? :
‘the polar equation to the required locus.
|
_ 43, Two given circles are traced upon a plane and a line
is drawn touching one and cutting the other in two points at
which tangents are applied to the latter circle; to find the locus
of the intersection of the tangents.
“0P=r, POO! =6, 00 =0, 0Q =a, O'@ =a! (fig. 69),
a
then. ¢ cos 0: =.0ON oe
a?
LF = ——____
ccos@—a!’”
‘he equation to a conic section of which O is a focus.
44. Having given the base and altitude of a triangle, to
ind the locus of the centre of the inscribed circle.
SC = CH =c=half given base (fig. 41), PN =a the
tiven altitude, O the centre of the inscribed circle, CM = 2,
WO = y its co-ordinates, CN = a’; then
a S’ a H
an S! = p> tan—= pe tan H = ——, tan— = 4 ;
C+a@ 2 C+a@ C—Z 2 C— x
a 2y (c+2) a 2y(c—a) |
ee c+a (c+a)—x¥’ sn" = (c- 2)? -y¥?’
12
|
therefore inverting and adding in order to eliminate a’, we get |
the required equation, which is of the third degree,
178
— —
45. Having given the base of a triangle, and the sum of
the other two sides, to find the locus of the centre of the
inscribed circle.
SH the given base = 2ae, SP + PH =2a, and SO= ry
HSO=8, (fig. 41), polar co-ordinates of the describing point O 5)
then area of triangle = 4 perimeter x radius of inseribed circle
=a(1+e)rsin@,
a’e(1 — e) sin 20
1 —ecos 20
r
os
also area of triangle = SP sin20.ae=
2ae (1 —e)cos@
1— ecos20
oie de =
the equation to a coni¢ section of which S'H is the major axis.
46. Two focal distances of a conic section include a con-
stant angle 2, and one of them is produced to meet the tangent
at the extremity of the other, to find the locus of the point ol
intersection.
B.
ST =r, 2AST=0, PSQ=[; (fig. 30), then :
a (1 — e’) :
= of = —~—_———._ (Art. 12
ASP = 23+0, and , AVG isd GON (Ar re}
the equation to a conic section with focus S'; and which 1s at
ellipse, hyperbola, or parabola, according as cos 3 >, <,or =@
If we draw another focal distance SP’ inclined at angle f
to SQ, then the tangent at P’ will pass through 7’ (Art. 128):
therefore the preceding is also the solution of the problem t
find the locus of the intersection of tangents to an ellipse a
the extremities of two focal distances that include a constan
angle 2/3. |
179
Also, if T'’P be produced to 7” so that zPS'7” — PST,
then 7" is a point in the locus of TZ’; from whence it follows
that the chords of a conic section whose eccentricity = e, that
_subtend an angle 2 at the focus, will be tangents to an-
other conic section having the same focus whose eccentricity
| = e cos f.
47. In an ellipse, if two focal distances r, 7, include an
angle = 28, and 7’ be the intersection of tangents at their
extremities, then
b? ry’
A Ae ae SET PS ya sere
6° — rr’ sin? 3
:
1— Z L 1 Sa a
We have Grae) = 1+ ecos@, eee +ecos @,
: r
: ‘a(1-e’) ; ;
‘and — go = 0088 +ecos AST =cos (6'— 6) +e cosh (0+6’) ;
‘between which equations if we eliminate @ and 6’, we shall
obtain the above result; which in the case of the parabola
becomes SJ? = 77’ agreeably to Art. 83.
48. To find the equation to the curve traced out by a
‘point in the perimeter of a circle which rolls upon another
equal circle.
Let 4’ be the describing point, at first in contact with A,
and 44’ the curve traced out, (tig. 70) ; C, C’ the centres of
the circles; join 44’, CC’, and let
AA =r, AAE=0, AC=a.
AA’ is manifestly parallel to CC’, draw DA’ parallel to
AC, and therefore = AC;
then 44’ = CD=CC' ~ LiGe
or 7 = 2a — 2acos@, the polar equation.
Or if AR = x, RA' = y, be the rectangular co-ordinates of A’,
i
o“+y’=2a (1-—* ,
Sairy Warns)
oe +y=2 a(r/ x +y” — 2).
iii
10M a
Hf:
im
ny
Hi
it
‘|
i
a PA na
SSS SS SS Ss SS
ee
180
49. ‘To find the equation to the curve described by a
point in the perimeter of a circle which rolls within another
circle of four times its radius.
P the describing point, at first in contact with A (fig. 70),
and AP the curve traced out;
CN=a, PN=y, CA=4a, QO=a4,
O being the centre of the rolling circle,
perpendicular to AC,
LPOM=7- (= -) —419=— — 305
ro)
~
w= CM + Pn =8acos0 + acos 30 = 4a(cos 6),
y = OM — On= Sasin 8 — asin 30 = 4a(sin ay
4 \ 4
(eyo
Aa Aa
i Suppose C A = 2a, the radius of the rolling circle equal a,
circle, but at a distance c from its centre ; then
LPOM=}47-90, «. v= (a+) cos@, y=(a—c)sin@;
x” 2
rei le es ee
(a+c) (a-cy)’
the equation to an ellipse; except a = ¢ when the equation is
y = 0, and the locus is the axis of a.
1,
50. Ifa triangle be inscribed in a Conic Section, and
each side be produced to meet the tangent at the opposite
angle, the three points of intersection will lie in a straight line.
If we take C for the origin, and AC =b, BC =a, for
axes, the equation to the conic section must be of the form
a+my +ney—ax—mby=0.
The equation to the tangent at 4 is
nb—a mb°
vw: and when y=0, v=
410 Oia ei yi nb—a’
y-b=-
ACD = 0; therefore QOP = 40, and consequently, OM being
and the point P to be not in the circumference of the rolling
181
which determines one of the points. The equation to the
tangent at B is y= — (vw -—a), and when w =0,
na—mb
y = which determine a second point. The equation
na — mb
— ax : :
- and equation to AB is
to the tangent at C is y=
ve Lit: ; mab?
say en: and for their intersection « = —__—. ,
a 6b mb? — a?
° a” e e °
y= rap aniatas the co-ordinates of the third point. Now the
mb? — a?
equation to the line joining the first and second point is
na—-mb nb—a
nae OE e=1,
a” m b*
and this is evidently satisfied by the co-ordinates of the third
point.
51. An ellipse being referred to conjugate diameters, if
with the co-ordinates of any point as conjugate semi-diameters
a second ellipse be described, it will be touched by the chord
of the former that joins the extremities of the diameters.
The equation to the interior ellipse will be
9 9
a
es e e h?
= 1, with condition = + 5 =1;
ee ® Mee
aX ~
We
‘and for the intersection of this ellipse with the chord
v Bare Bi 2 i
at pmb we have G+ = (1-*) = 1, or (ax —h*)’=0,
a a
Ween le
which shews that the chord is a tangent at a point for which
h?
v= —.
a
52. The chords joining the extremities of conjugate dia-
meters of an ellipse will all touch in their middle points a
similarly situated ellipse with axes ar/ 2, b a/ 2.
i
Se “<< *
182
The equation to the inner ellipse and to the chord, re- |
ferred to a pair of conjugate diameters of the outer ellipse as
axes, will be respectively,
~
et Tae
3/8
+
IS
See
=
Q|s
Cte
—
we
FY Ce Qa a
2(—+1-2—+—, =: 1 OP bee = =O 5
A he a 7 Pas
therefore for their intersection we have,
which shews that the chord is a tangent in its middle point.
53. To find the locus of the intersection of two tangents
to an ellipse applied at the extremities of a chord which always
touches a concentric and similarly situated ellipse.
Let a, 6, be the semi-axes of the exterior ellipse, a’, 0’,
those of the interior ; and (h, &) the point through which two
tangents to the former pass; then the equation to the chord
joining the points of contact is
he ky :
a” st e
’ id
which must be identical with —- - = 1,
ae 2
the equation to a line touching the interior ellipse at a point
(a, y’); therefore |
hence since (=) *
On.
i. = 1, the equation to the required
ye ;
locus is (=F) + = ‘) - = 1, representing a similarly situated
2
a
ellipse with semi-axes —, =
ab
54. An ellipse whose centre C is given touches a fixed —
straight line PQY in a given point P; to find the locus of
either focus S.
|
183
Let PQ=h, CQ=k (fiz. 105) be the co-ordinates of
C, which are known, and PY =a, §Y = y those of §; then
since CY is parallel to PH, zCYQ = SPY,
k y k oh
a & = - OFr.-+-— = 1,
ec-h @& Yi a
the equation to a hyperbola.
55. If an ellipse and hyperbola have the same foci, the
locus of the intersection of tangents to them, at right angles
_to one another, is a circle.
The equation to a line touching the ellipse is
Y-ML= a/b? + ma’,
: e s 1 .) e ry
and changing m into ——, and a’, 6°, into a?, —b', the
m
equation to a line touching the hyperbola, and at right angles
to the above, is
my +e =/— mb? + a?;
where, since the curves have the same foci and centre,
NOC = 0 — ft =a" fp?
Adding the squares of these two equations,
(2? +. y’) (1+ m?) = 6? + @ 44m? (a? ie b’?) xi (b? ve a’) (1 4. m*),
or @ +y=h 4+ a”,
the equation to a circle passing through the four points of
intersection of the curves.
56. To find the locus of the centres of the ellipses in-
scribed in a given quadrilateral.
Take lines through one of the angles of the quadrilateral
parallel to two sides for the axes of the co-ordinates; and let
w=h, y=k, y=mex, y=na, be the equations to the four
sides; then the conditions for these lines, respectively, being
tangents to the ellipse, supposing its equation to be
ay’ +bay +c2#°+dy+er+1=0,
Se —
SS
———$ =
a
A ae Es eS as a SS SSS SSS SS : Bee SSS
ee Se eee ie ee —— ————— a: = ee
i
ae eS
184
(found by making the two points coincide in which each cuts
y s P
the ellipse) are
4a (ch? + eh +1) = (bh +d)’,
4c (ak? + dk +1) = (bk + e)’,
4 (am* + bm-+e) = (dm +e)’,
4(an° + bn +c) = (dn + e)’.
But if 2’, y’, be the co-ordinates of the centre of the
ellipse, the two former become (Art. 223)
(4ac — 0°) (h? — 2ha’) + 4a-a =0,
(4ac — b’) (KP — 2ky') + 4c —e& =0;
and eliminating 6 between the two latter, we get
(4a -—d’) mn = 4c -e*,
* h? —2ky =mn (hl? — ghar)
the required equation; which represents the line joining the
middle points of the diagonals of the quadrilateral. -
That the locus of the centres would pass through the
middle points of the three diagonals might have been fore-
seen; because each of the diagonals may be regarded as the
transverse axis of an evanescent ellipse touching the four
sides of the quadrilateral.
If one of: the angles become equal to two right angles,
the ellipses are inscribed in a triangle, touching its base in
a given point; and their centres lie in the line joining the
middle point of the base, with the middle point of the line
drawn from the vertical angle to the common point of contact.
57. Ina given triangle to inscribe an ellipse of given
area, and touching one of the sides in a given point.
Let P be the given point in the side BC (fig. 106) and M
the middle point of BC; draw MS bisecting AP, and AQ
cutting off QC = BP; then the centres of all the ellipses that
185
can be inscribed in the triangle and touch BC in P, lie in
sequently the loci of D and O are as asserted. Let DQ = z,
AQ=k, MC=a, BP =c; then
DIT leas
RC tk
and OP..sin OPM = 4 DQ.sin DQP = 4 % sin w, suppose,
Cc
or DT => (k-2),
Now (area)? = 7’ sin°OPM.PC.DT. OP?
mT’ sin’w(2ac—c*) ,
cea Re RAIS 9 all
Ak ( 2)
If the area of the ellipse equals the area of a circle radius r,
then (2ac¢ —c’) sin’ wx” (4 — x) = 4kr*
is the equation for finding x. For the greatest inscribed ellipse
that touches BC in P we must evidently have
x= 2k, or MO = 280.
1 929.09
Then 7* = za k* sin” w (2ae — c’),
which is a maximum by the variation of ¢ when ec = a, or P
coincides with M. Hence the greatest of all the inscribed
ellipses touches each side in its middle point, and has its centre
coincident with the centre of gravity of the triangle,
58. In the equation ay’? + bay +ca’?+dy+ev+1=0,
suppose 6 to assume different values, all the other coefficients
remaining unchanged; then (1) the conic sections which it
represents are in general all described about the same quad-
rilateral; (2) the locus of their centres is another conic section,
whose equation is 2ay’+ dy = 2ca* + ea; and (3) the centre
186
of this last conic section is in the middle point of the line
joining the bisections of the diagonals of the quadrilateral.
It is evident that the four points in which the curve cuts
the co-ordinate axes are independent of b. If h, k, be the
co-ordinates of its centre, then
2ak+bh+d=0, 2ch+bk+e=0,
between which eliminating 6, we find the locus of the centre to
be the conic section, whose equation is
2ak? +kd = 2ch’ + he.
If h’ and k’ be the co-ordinates of the centre of this curve,
, e d
h’ = —-—=1(7,4+%,), k= - a 1 (y, + Y2), (Art. 240)
which are the co-ordinates of the middle point of the line |
joining the bisections of the diagonals. It may be shewn that !
the curve passes through the intersection of the diagonals, and
also through the points of intersection of each pair of opposite |
cH sides. |
|
|
59. To find the locus of the point which is the inter-
section of three normals to an ellipse. :
The equation to the normal at a point (#’, y’) of am.
ellipse is
a y'
Y si y 7 be a (wv 7 a),
or ya \/1 —@ = (a — Ba’) fa? — &”. ¥
Let h, & be co-ordinates of a given point through which
the normal passes, then |
Kw? (1 — e&) = (h—ea')? (a? — @”)... +02 00-(1)
is the equation for determining a’, the abscissa of the point
in the ellipse; and as this equation is of the form :
ex! a &c. sae a? h? == 0,
187
it has two or four possible roots; and consequently through
‘the point (h, k) in general either four or two normals can
be drawn. If two BP the possible roots become equal (which
jcan only happen in the case of four real roots), then three
normals will pass through the point (h, k); in that case
ithe derived equation
Kea! (1 —e*) = —& (h -— ea’) (@ -— w”) — or (h- Ca’?
has one of them. Dividing this by (1), we find
| t ;
1 e & ee ee
SG = Se Or he? = C* ar? s
a h-@uv a—«x?*
ha\s _, Byes
ae = ee satisfies equation (1), and substituting we get
Ke
k* (1 — e’)3 +h® = (ae)?
for the equation of condition that (1) may have equal roots,
and as often as h and k satisfy this equation, three normals to
the ellipse will pass through the point (hk, k). The above is
consequently the equation to the locus of the intersection of
three normals to an ellipse; and coincides, as might have been
foreseen, with the equation to the evolute.
60. If the tangents at the extremities of any diameter of
mn ellipse DD, fs intersected by the tangent at any other
ooint, in 7, 7"; then DT. D'T" = CP’.
The equation to the tangent at Q (fig. 47) is
, 4
ve yy
BR pals / ? °
md making y = b’ and — 0b’, successively, we get
av’ , av ,
DTH ee MR THO RN Sey, pt
OF b a” b
, , a? y” trp / 9
Oy TOD LET Bd bee 7 = j2? or DT .DT Sa =. P*,
a” 2
61. The greatest pb that can be inscribed in a quad-
‘lateral that has two sides h, k, parallel to one another, will
188
touch those parallels in their middle points; and its area will
=f 1 +14/hk, l being the perpendicular distance of the par]
rallels from one Ethan
Join the two points of contact of one of the incr
ellipses with the parallel sides by the line DD’, (fig. 100)
bisect it in C, and through C draw PP’ parallel to D7’; then
C is the centre, and PP’, DD’, are conjugate diameters
of one of the ellipses inscribed in the trapezium; and if
CP=a, CD=b, DT =c, DT =d, @=cd=(h-@
x(k —d) (Prob. 60); .. kec=h (k-—d). Now (area)? of ellipse
«a «(k—-d)da«tk® —- (1k -d)’; therefore, for maximum
area, d= 4h, naa c=th; and maximum area = 7a.4)
= La yia
If h =k, the trapezium becomes a parallelogram, and the
greatest ellipse = 4 x area of parallelogram.
62. In a given parallelogram to inscribe an ellipse of
given eccentricity.
Every ellipse must have its centre in the intersection of
the diagonals; and as in the preceding Problem, if Q be the
middle point of the side RT’, and CQ=1, QT =k, QD= a
PCQ = 0, PCD =.0, CP =a, CD = 6,. then
a= DT.DT =k — 2,
P=? +2lscos8 +2", bsinw= sin.
If therefore a and £ be the semi-axes of the ellipse
a’ + BP =k? +2? + 2lzcos8,
af = lsinO\/k? — 3°;
+k
. Lsin @ te + P) sa + 21 cos@. Sailnet,
Boa VJ kt — x? J ke? — 2°
the equation for determining x, since Fa Ste is given,
189
Since the ratio of 3 to a is zero when the ellipse coincides
——-
with either of the diagonals, it must admit of a maximum ;
and the corresponding value of x may be obtained from the
—
:
;
2quation
(Ao + P) x + 21k? cos8=0, (1)
which determines the ellipse that approaches nearest to a circle
of all those that can be inscribed in a given parallelogram.
If the parallelogram be equilateral, and if PD, CT, be
‘coined, then CT’ bisects both the chord PD, and the angle 7’
>]
and therefore bisects the chord perpendicularly ; : therefore Cr
' which gives a.
san Axis of the curve; and if CY be a perpendicular from C
m RY, since YCT =1 (7-8),
CY=ksn0=av/1 — e’ cos’ 4 0,
If e=0, QD evidently equals & cos @ which
Figrees with (1) when J =k.
63. ‘To inscribe an ellipse in a semi-circle, which shall
-iave a given major axis parallel to the diameter of the semi-
eirele.
CN =a, NP =y, the co-ordinates of P (fig. 107); then
-»ecause the normal at P passes through the extremity of the
-ninor axis,
y= 7 ore but (y +6)? +a? = BP? =r’,
a®*—b
a’
or (y+ b)'+ OU —y) ar, .. ata (a? — 8),
orb =e
Hence in order that the area may be the greatest possible,
ve must have ab a maximum, or a? a/v — qa’ a maximum ;
—
9
ae + WE , and greatest area =
2
2
9
Qarr-
aV3
Sa ea So Se
190
64: To find the locus of the middle point of a straight line
that always has its extremities on the circumferences of two
equal circles given in position.
A, D, (fig. 108), the centres of the circles, O the middle
point between them the orign; ON=a, NP=y the co-
ordinates of P the middle point of BC; 2’, y'; wv’, y”, the
co-ordinates of B and C; BC =2c, O4=OD=b, AB = CD
=a; then
2yay +y’s
2v=2 —2',
Ae?= (y’ —y")? + (a +2’,
a =y” + (a — 5)’,
aay”? + (a —b)’,
between which five equations we have to eliminate a’, y,
/
wv’, y"; subtracting (4) and (5) and reducing by means of
(1) and (2), we get
(6) yy -y") + 0(a' +0") =2be.
Again adding (4) and (5) and doubling
2(y? +9?) +2 (a? + a") — 4b (a! + 2”) = 4(a2 - B),
but (y' + 9)’ + (#’ — a”)? = 4 (a? + 7)
from (1) and (2); therefore, subtracting, and reducing by (3),
(7) e+e’ =—-(P+ce—a'+a*%+y"),
1
b
2
be «x
also from (6) y/ —y" = —— — - (a +2”
Ms a
L
be ve. 2B 2. @?);
hein ail dc Lae
. substituting in (3), and reducing, we get for the equa-_
tion required
o 9 9
~
P(e +y+eO-7 -8Y 4g (4+ yt -27+ bY = 4 ey’.
191
65. ‘To find the locus of the intersection of two normals
to an ellipse at right angles to one another.
Let m be the tangent of the angle which a normal to
the ellipse makes with the major axis, then its equation
ds (Art. 125),
(y — ma) fa? + mb? + (a? — B°) m = 0,
'and changing m into , the equation to a normal per-
| pendicular to it, is
(my + 2x) / ma? +B? — (a” —b’) m= 0,
-and we have to eliminate m between these equations.
We get by addition,
a+mh (my+ay
mar+b? (y—ma)
C1);
a+ atm? (a? + m°b*) (ma +8) ,
ety (myteP (abn
a’ +b? 1\2
Pe -(a-by= + bP + ab! (m=) 2).
sng (a BP = (a +8 +)" ©
But from (1) we get
a-P vy AMLY
oe ° = Q ” =f . » 9 ~ 3
a+b at ty? (a + y’) (1 —m’*)
1 Quy (a? + b*)
—-— m= - noted
m ay — Pa?’
which value substituted in (2) after reduction gives-the re-
quired equation
(a” ny 5°)” ee oti al
(a at b°) (v” + y’) His a*y? — ba? :
66. ‘To find the locus of a point from which if four
normals be drawn to a curve of the second order, the sum of
their squares shall be constant.
‘|
192
- Let y? + na*=c?® (1) be the equation to the curve, and
(a,b) the point whose locus is required; the equation to
!
the normal through it is
nba
y
—-b=—(«#- et) ees
J ND Gir a (n—l)@+a
nbx 2 Wer
7. —— eS 0S
(n-l)e%+a
Qa n (a> + 26*)—(n =1) c
2° ( ag ) a* + &e. = 0.
4
or @ ¢ -
weet n(n — 1)”
Similarly,
2nb n (a* + nb?) —(n —- 1)? ec
vey? u Jost Be 7 te
(2 — 1)?
24n(a* + nb?) —-(n- 1)?
(m — 1)°
3 (-2by) = —26 (=) enti
n—1 n—1
4b? = 4b’,
3 (wal +(y— by"
as 6
o
jan 2) = 4R* a constant ;
n
An —2 2n — 4 ]
or a® ( jae. ) =4K?-20 (1+-);
nit reek 1 n
or of. @m—1) +0. (w=2) = (n= 1) 2B ( +7)}.
193
67. If an ellipse be inscribed in a quadrilateral, the lines
| joining the extremities of either diagonal with the points of
contact will intersect in the other diagonal.
In fig. 109, because the sides of the triangle KAC are cut
by a straight line in the points N, J, M,
| EN WEN GA Te
KM Gl aM’
KM CR MD
bites 5 epee ee
a au SIT VERE
because the ellipse is inscribed in the triangle KCD,
“ CR.MD.AI=DR.AM. IC,
which proves that if CM and AR be Joined they will intersect
am DI. Also, because the quadrilateral is circumscribed about
he ellipse,
Aly DR.AM AL.BN
IC CR.DM” BL.NC’
or BL.NC.AI= AL.BN. IC,
: thich proves that AN, CL, BD, intersect in a point.
| 68. If an ellipse be inscribed in a quadrilateral, the line
dining its points of contact with two opposite sides passes
orough the intersection of the diagonals.
| Let K the point in which the opposite sides BC, AD in-
“sect, be taken for the origin (fig. 109). KA=a, KB= b,
=c, KD=d; and KN =k, KM =h, M and N being
he points of contact in AD, BC; then the equations to BD,
'C, and NM are, respectively,
: v Y v Y Y
Soe AT OY ecm +-=1;
: Tema CE To kepeais
:
id therefore the condition for their passing through a point is
ee
194
Now the equation to the ellipse is (Art. 228),
(; 1) = (7 1) may =1
h i ig J
and for its intersection with the line CD whose equation is
d
= 1, we have
c
a 1) (bi ( k 1) i ( “) ;
—_— —{—-4--—- mec —-—j)=1;
E te \at ce d
which must be a perfect square since CD is a tangent,
m 1
2 2G 1) (
Pee h ~ ab
. “ 4 e ° ° e,e :
(since — + a 1 is also a tangent), which is the condition (1),
a
69. To find the area of an ellipse inscribed in a given
quadrilateral, and touching one of the sides in a given point.
As in the preceding Problem, taking two of the sides ol
the quadrilateral for the co-ordinate axes, the equation to the
ellipse will be
i 1) Y Get Inky?
a + (2-1) +270y=1, (ihk)’ <1;
1\ )2 2
+o yak(r ho) kle- 2 (e+ 7) {(;-18) af.
v v
Hence when the ordinate becomes a tangent at N’, Ww
have
oh
yay tee
“é hkl+1
195
Therefore when «= 1K, the radical in the value of y
1—hkil\2
——]} = OQ the seni-
mT cal Q e semi
assumes its greatest value = (
yt .
diameter conjugate to NN’:
;
(1 -hkil)2
(1+ hkl)2
area of ellipse = rhk sinw
| AI BI BL :
: Now let To 7” Tw ane L being the given
loint where the ellipse is to touch AB;
then, since BL.MA.DI= AL.MD. BI,
AM wm AM mM
=—, and = .
MD 3 AD m+e
But KA ABAC_ BI AC m (2 +1)
| LED TABDOCPIRDS 1C8 m+1
KA m(n+1) AM 1—-mn
= su hencepest7 ‘
AD 1-mn KA (n+1)(m+32)
AM a I1—mn
ang == 1-- = ;
KM h 1+m+sz(14+n)
(1-—mn) x ;
l+m+z(1+n)’
- 1+m)3s°+ (1+m) 2?
: area = Lad sinw (1 —mn) 21+”) + +m) eth
3m (” +1) +(m +1) nzh3-
consequently 1 —
b
k
70. In a given quadrilateral to inscribe an ellipse whose
a shall be the greatest possible.
The expression for the area in the preceding Problem
lishes when x = 0, and when = © ; and also when
132
196
(1+n)#+1+4+m=0, which gives AM = — AK; corresponding
to which values of x the ellipse becomes coincident with the
diagonals BD, AC; and with the line joining K, and the
point in which 4B, CD intersect. The area of the ellipse
will consequently be a maximum for some value of x between
zero and infinity given by the equation
: (== me) m.
re 2 eee es
n+I1 n m+)
The ratio v of CR to RD must be the same function of
1 1 ; , :
" and — that ¢ is of m and m, and is therefore given by the
m
”
° 9 mnt+l m+i m t
equation v + 2v |—. —- —— =0; which shews
mm+il n+i n )
that the negative value of » taken positively is the value of
the ratio of CR to RD.
|
71. Ina given quadrilateral to inscribe an ellipse whose
axes shall have a given ratio.
By Prob. 69, since the equation to the diameter NN’ is
k—y=k’lx, we have 7
h?
ON = ray ee ee eps
and ete Hye mes
ba i |
at 65 h? +k? — 2h?k?l cos
a a
(LE RED: ane
. Q=hkd!
| and afp = hk s1n @w hE 9 j
a and B denoting the semi-axes of the ellipse; also let |
denote their ratio, ;
1 |
.. sin (v + -) hk {1 — (hkl) tt=h? + k? - 2h? Kl cosws
Ss 2y
Bie ETA Te? PCa ay Es as
(m +1) (nm +1) (m+ 2) (14 72)
Sk 1—-mn k (1—mn) x
since -—-1= — Se ach se oan Bete CRS at sf
a (m + 1) (m + 2) b (m + 1) (1 + nz)
hk @ mote + ne
also - = —. =?
| kee bee 1a or +
and upon substituting these values there arises an equation of
the fourth degree for determining x.
72. To find the locus of the middle points of a system
of parallel straight lines, each of which joins two points in two
given curves.
Let f(v, y)=0, d(#,y)=0, y=ma+e, be the equa-
‘ions to the two curves, and to one of the chords; transfer
he origin to (a’, y’) the middle point of the chord; then the
‘quations to the curves become
f+, ¥Y+y=0, P(e +a, y+y) =0,
md the equation to the chord y= mwa; and if in the former
ve substitute ma for y, the resulting equations will give values
or #, being the abscissze of the points of intersection of the
hord and the curves; and if + satisfy one of the equations,
he other must be satisfied by —; therefore the equation to
he locus of the middle points of the chords will result: from
liminating x between
f(@ +a, y+ mex) =0 and d(a' —«, y’- mz) =0.
Suppose the curves to be a hyperbola and its conjugate ;
he result of the elimination will be found to be
4 0'b' (a?m? — b*) = (a? my — Ba’)? (Ba’? — a?y’”).
If the asymptotes be taken for axes, the result will be
ay’ (ma! + y')? + mc =0.
73. ‘To find the locus of the vertex of a triangle, upon a
iven base, and having its vertical angle bisected by a line
arallel to a given line.
Take the given line for the axis of y, the middle point
‘the base (2a) for the origin, and let the angle between them
a; then AM =a coseca, RN =a cosa; (fig. 110) ;
198
|
heat a+acosecca BM BP PR _¥~acoesa |
at - = = = a |
Pn £Y ce meoseca MIC MERC 0 BS. | Yy + @ COS a’
xy = 4a’. sin 2a.
74. 'To find the locus of the intersection of the tangent :
to a given curve, and the perpendicular let fall upon it
from a given point. :
Let y —y’ = tan a (w — 2’) be the equation to the tangent
to a curve at a point (a’, y’); then tana = f(a’, y’) is known:
from the nature of the curve; and the equation to a perpen-|
dicular to the tangent from a point (h, k) is (y—k) tana+a@
—h=0; between which three equations, and the equation to
the curve if a’, y’, and tana be eliminated, we shall obtain the:
required relation between # and y.
Thus the equations to the tangent to a parabola and to a
line perpendicular to it from (h, &) being :
a
il
mM m
the equation to the locus of their intersection is
a(y—k)’+a(a—-h)’+y(y—k) (@-h) =
which, if h=k=0, becomes y’ (a + x) +. v*?=0 the equation
to the Cissoid of Diocles; and if k =0, h = a, it becomes
w fy? + (a@—a)} =0, or x =0,
the equation to the tangent at the vertex.
Similarly, the equation to the tangent to the ellipse being'
Y-Me= / b? + m?a®, the locus of its intersection with a
perpendicular let fall from a point (4, &) has for equation |
y(y—k) +a (a—h) = 4a? (w@—h)?+ B (y —k)*}2;
which, if the perpendicular be dropped from the centre, be-
comes |
(a? + o?)? = aa? + By’,
which agrees with the polar equation already found (Art. 135)
r” = a? (1 — e” sin’).
199
In the case of the hyperbola, changing b? into — 4’, the
equation is
(a? + y?)? = oa ~ 82 Ys
which if 6= a becomes
(a* +")? = a? (a — y’),
representing a curve called the Lemniscata of Bernouilli, and
_ whose polar equation is 7°= a?cos20. If the perpendicular
be let fall from the vertex of a rectangular hyperbola, the
equation is w+ y= a(# — V/ x — y’).
75. If a curve roll upon an equal one, similar points
‘being always in contact, to find the locus of any given point
in the rolling curve.
| Let AP be the fixed, and A’P the rolling curve (fig. 71),
8” the describing point, and |S a point similarly situated in the
fixed curve. Join S'S" meeting the common tangent at P in
-Y; also join SP, S’P. Then because the points in contact
‘are similar, S'P, §’P are equal and equally inclined to the
common tangent PY; therefore PY bisects S18" at right angles,
and therefore the locus of JS” is similar to that of Y, the foot
of the perpendicular from § upon the tangent to the fixed
curve, and S"P is always a normal to the locus of 5S”.
If therefore y= f(a) be the equation to the locus of Y,
ES (5) is the equation to the locus of §’. Hence if the
curves are equal parabolas, and JS’ the focus, its locus will
‘be'’a straight line; if § be the vertex, its locus will be the
Cissoid of Diocles ; if the curves are ellipses, and |S the focus,
fits locus will be a circle; if § be the centre, the equation to
its locus will be
et yar Sara + b*y’.
; au (e@— 3a)
76. ‘To trace the curve 7? & - iy a
C— 44
e a 4
When vw =0, y =0, and limit of Ss a Cs ; therefore at the
origin the curve cuts the axis of # at right angles. When
xv = 3a, y=0, and limit of , when & = 34, is infinity, ©
a
v
therefore the curve again cuts the axis at right angles at a_
distance 3a from the origin. For values of w between 3a and |
4a, y is impossible, and there is no curve; when # = 4a, y is |
infinite; and when 2 is very large, the relation between x and |
y becomes
3 4a\—} a Aa
y= aa ( 1 —-— = ax UD ESS grees 9
L v av
“. y’=a(x+ a) is the equation to the parabolic asymptote, —
above which the curve lies. ‘There is a maximum: ordinate |
2
By taking the limiting value of , it will be found that the
: 3a ee J a |
diameter of curvature at 4 = Tae and similarly at C it will |
be found to equal 3a. ‘There is no part of the curve cor-
responding to negative values of w; and the axis of a is an |
Axis of the curve. Hence the curve is such as is represented
in fig. 111, the dotted line being intended for the parabolic |
asymptote.
77. 'To trace the curve
(y? — 2°) (w — 1) (w - 8) = 2 fy’ + a(w — 2)}?.
Solving the equation relative to y*, we find
ay -§@-Wa- ps8 @- nV Cor Ddoosd,
Hence # must lie between — 51, and 3; and as the rational
part of 2y° = 3(# +4) (25), — 2) nearly, when w = — 5}, the
ordinate is real, and is a tangent to the curve; when w=0,
2y°= 3 or = 0; when #=1, y’=1; and when #=3, we
get a real value for the last ordinate, touching the curve;
which consequently is that represented in fig. 112, having
three true double points, and four double tangents, i. e.
straight lines touching it in two points.
201
7 — a3
78. To trace the curve y= tae
eC—2a
a
When v=0, y= +—~=;
~
when wv = a, the curve cuts the axis of a at right angles; from
“w= a to &= 2a, y remains impossible; when # = 2a, y is in-
finite; and when #w and y become very great, the relation |
between them is i
Loin
a’ eh ies Cod (te Sue it
a alee. Re lgbitae act om, |p cel to tegas Hy
2 2x” BE Sag" Gee Za ii
. + y=a+a is the equation to the asymptotes above which
the curve lies; also when a is negative, y perpetually in-
creases, and there are two infinite branches; therefore the
curve is such as is represented in (fig. 113), the co-ordinates
of the points P, P’ where it cuts the asymptotes being
a ee
V=—-—-, Y= mae @
3 3
79. The corner of a page is turned down so that the |
triangle is of a constant area, a?; the locus of the angular
point is a lemniscata whose equation is 7” =a? sin 20.
80. If two circles be inscribed in another circle touching i
one another, then the area of the circle whose diameter is
their common tangent, will equal the area between the greater
semicircle and the two smaller ones.
81. The equation y* + 2a?xy — v* = 0 expressed by Polar i}
co-ordinates is 7” = a? tan 20.
82. Of the three squares that can be inscribed in an acute- iH
angled triangle, the greatest is that which has two angles in hh
the least side.
83. A parabola is bounded by an ordinate perpendicular i
to its axis, whose length is 6, that of the portion of the
Ss
202
axis cut off being a; D, d are the diameters of the circum-
scribed and inscribed circles, then D+d=a + b.
84. In PG the normal to an ellipse, a point Q is taken
such that PQ = CD, shew that Q traces out a circle.
85. Two conjugate diameters are produced to intersect
the same directrix of an ellipse, and from the point of in-
tersection of each, a perpendicular is drawn to the other ;
these perpendiculars will intersect in the nearer focus.
86. If a pair of conjugate diameters of an ellipse when
produced, be asymptotes to a hyperbola, the point of the
hyperbola at which the tangent will also touch the ellipse,
lies in an ellipse similar to the original one.
87. If two given circles touch one another internally, and
a series of circles whose radii are 7,, 7,, &c. be described
between them touching one another; and if P,, P,, &c. be
the perpendiculars dropped from their centres upon the com-
n
mon diameter of the given circles, then —
Tn
88. If a, b, r, be respectively the radii of the given
circles, and of the first circle in the series, prove that the
radius of the (7+1)™ circle will be
ab(a—b)r
abr + jn (a — b) fr &/ab(a—b—r)t?
89. If two circles touch one another, the radius of any
circle touching them both bears an invariable ratio to the
perpendicular from its centre upon their common tangent. .
90. If the length of the axis of an oblique cone be
equal to the radius of its base, every section perpendicular
to the axis will be a circle.
91. If an ellipse be moved between two straight lines at
right angles to one another, to shew that the centre will
describe a circle, and to find the locus of any given point in
the axis.
203
92. To find the equation to the conic section described
with focus (A, &) and directrix y = ma + e.
93. If SY, HZ be perpendiculars from the foci upon
the tangent at any point P of an ellipse; then §Z and
fITY will intersect in the middle point of the normal at P,
and the locus of their intersection will be an ellipse with
a(1 +e’) and a\/1 —e? for axes,
94. If a parabola be moved between two straight lines
at right angles to one another, the equation to the locus of its
. 4 2 42
vertex will be ay? + y3a3 = a’,
95. The area between two normals to a parabola at the
a 200 :
extremities of a focal chord, and the curve, = —_._—_, 9 being
3 sin® 20
the inclination of one of the normals to the axis.
96. The sum of the squares of the normals to an ellipse
drawn at the extremities of conjugate semi-diameters
= a’ (e® — 1) (e? — 2).
97. Find the locus of the vertex of a triangle whose
base is constant, and likewise the product of the perpendi-
culars dropped from the extremities of the base upon the
line bisecting the vertical angle.
98. If P be a point in a hyperbola, whose ordinate
= BC \/ SC, and CY be a perpendicular from the centre
upon the tangent at P, then PY = SC.
99. If the opposite sides of a hexagon inscribed in a
conic section be produced to meet, the three points of inter-
section will lie ina straight line.
In fig. 114, draw any diagonal MM’, and let the pairs of
opposite sides which pass through its extremities meet in CG;
B; and taking the line CB for the axis of #, let the equation
to the conic section be
Sea as wes SS = ———e Fr -
a Sais we eX Sees
Woe SS eee
204:
ay’+bay+cx°+dy+ex+f=0 (1),
and let the equations to the sides M’M, MN, NN’, N'M’ of
one of the quadrilaterals into which MM’ divides the hexagon,
be respectively
qu+qy=1,
se+sy=l.
pxe+py=1,
ya+ry =1,
Now the equation to the conic section is satisfied by all
such values of w and y as jointly satisfy the equations to any
two adjacent sides of the quadrilateral ; and therefore its equa-
tion, since it is of the second degree, must be of the form
m(px+p'y—1) (re+r y—1) +n(qa+q y—1)(sa+s'y—1)=0,
which compared with (1) gives
mpr+ngs=c, mt+n=f,
m(p+r)+n(q+s)+e=0.
Now if we suppose 4, B, C to be given points, and there-
fore p, g, s to be given quantities, these three equations deter-
mine m, m and 7; therefore D is a fixed point; which shews
that if three sides of a quadrilateral inscribed in a conic section
pass through three fixed points in a given straight line, the
remaining side also will pass through a fixed point in that
line. Consequently, since three sides of the quadrilateral
MOO'M' pass through the points 4, B, C, the remaining side
OO' must also pass through D; therefore the three intersections
of the opposite sides of the hexagon lie in a straight line.
If the hexagon be changed into a triangle, by supposing
every other side to become evanescent; and therefore to assume
the direction of a tangent to the conic section at. one of the
angular points of the triangle, we fall upon Prob. 50.
100. If two pairs of opposite sides of a hexagon inscribed
in aconic section be parallel to one another, the two remaining
sides shall also be parallel to one another.
Let MM’ be any diagonal of any hexagon inscribed in a
conic section having two pairs of opposite sides parallel to one
205
another, and as in the preceding Problem, let the equations to
the four sides of the quadrilateral M’MNN’ be
05 y+qu+q =0,
Y+pet+p
Ytra+?” =0, Ytsu +s =0;
then the equation to the conic section will be
m(ytpxtp’)(ytrot+r) tn(ytqu+g') ytsets’)=0,
which compared with (1) in the preceding Problem, gives
M+ =A, Mpr + NYS = C,
m(r+p)+n(s+q)=b.
These equations shew that if three of the quantities
p, q, 7, 8 be given, the fourth is also constant; i.e. if three
of the sides of any quadrilateral inscribed in a conic section
be parallel to three given straight lines, the remaining side is
also parallel to a fixed line. But if O’M’', OM be respectively
parallel to the lines to which MN, M’N’ are parallel, then the
position of OO" is determined from the above equations by
interchanging gq and. s which does not alter them; therefore
OO’ is parallel to the same line to which NN’ is parallel; or
the two remaining sides of the hexagon are parallel to one
another.
101. The three diagonals of a hexagon circumscribed
about a conic section intersect in a point.
Let aBrydex (fig. 115) be the angular points of a hexagon
circumscribed about a conic section; join the points of con-
tact by straight lines, so forming the inscribed hexagon
ABCDEK ; and produce its opposite sides to meet in P, Q, R.
Then if two ‘tangents were applied at the’points a’, \, in
which the diagonal ad meets the curve, they would intersect in
P; similarly the pairs of tangents applied at the points where
yx, Be, meet the curve, would respectively intersect in Q and
Rk; but P, Q, R, lie in a straight line; therefore the three
diagonals (since they are in the directions of chords joining the
points of contact of pairs of tangents drawn from points in a
straight line) must (Art. 50) pass through the same point.
SECTION XI.
ON CURVES OF THE THIRD AND FOURTH AND HIGHER ORDERS 5
AND ON THE SINGULAR POINTS OF CURVE LINES.
250. Iw this section we shall give some of the principal
results that have been obtained relative to the properties of
curves of the 3rd and 4th orders; and as Pliicker, to whom
the following investigations are chiefly due, has applied his
method to curves of the second order, as well as of higher
orders, we shall commence with that application ; both for
the sake of some new results to which it leads, and for the
purpose of making Pliicker’s general method more readily
understood.
251. ‘The general equation of the second degree
y +2Aavy+ Ba’?+2Cy+2De+H=0, (1),
provided s°+2Az-+B=0 has not equal roots, can alway
be transformed into
(y+an+b)(y+aue+b)+m=0, (2) ;
only when the auxiliary equation has imaginary roots, the
factors of the transformed equation are likewise imaginary,
but their product real.
As equation (2) is of the 2nd degree, and contains the
requisite number of independent constants, we may evidently
assume it to be identical with (1); and upon expanding and
equating coefficients, we find
ata=2A, ad=B;
so that —a, — a’, are the roots of 3? ++2Az+ B=0, and will
be real provided 4?- B>0. ‘To determine b, 6’, and m, we
have |
b+40'=2C, ab’ 4+ab=2D, bb'+m=E, (3)
207
therefore if a, a are real, these equations will evidently
_ furnish a single system of real values of 6, b’, and m. But if
a, @, are imaginary, i.e. if 4? — B<0, the equation
ab'+ab=2D
shews that b, b’ must also be conjugate imaginary roots of a
quadratic equation, and their product consequently real, and
therefore m real; and in this case the two factors of the trans-
formed equation are imaginary, but their product (which
equals — m) is real. Hence the proposed transformation can
be effected, and only in one way; none of the coefficients
being indeterminate, nor having more than one value.
252. Ifa and @’ are equal, then A?— B=0; and cqua-
tions (3) become inconsistent with one another, unless
Dieta AG.
when the two former of them become identical. Hence when
A’-— B=0,
the transformation (2) is impossible, and it may be replaced by
(y+axv+b)’?+m (y+cu4+d)=0,
in which one constant may be assumed at pleasure (since the
general equation, with the condition 4? — B = 0, contains but
four independent constants), and then all the others can be
determined from linear equations.
If from the two former of equations (3) we determine
b, b’, and substitute them in bb'+m=E, we get
m (A* — B) = D?’-2ADC + BC? + E (4? - B);
and if the second member of this equation vanish, the pro-
posed equation resolves itself into two factors of the first
degree, and represents two straight lines; if the second mem-
ber be negative at the same time that A? — B is negative, the
proposed equation cannot be satisfied by any real values of
we and y.
208
253. When the general equation to a curve of the second
order is put under the form
(y+ax+b) (y+a'xr +b’) +m =0,
its two real or imaginary asymptotes have for equations
y+an+b=0, yt+taurs+d'=0.
If with the equation of the second degree, which as
we have just shewn may be written pg + m= 0, where
p and q denote linear functions of # and y each containing
two constants, we combine the equation to any straight line,
we shall usually determine two points of intersection ; but if
° m .
we take the equation p = 0, we get — = 0, which can only be
7
satisfied by supposing # and y to be infinitely great; so that
p =O represents a straight line whose two intersections with
the curve are at an infinite distance, or it is the equation
to one of the asymptotes of the curve; and in the same way
it appears that g=0 is the equation to the other asymptote.
The curve is a hyperbola when the asymptotes are real, and
an ellipse in the contrary case; and instead of being deter-
mined by five constants as in the case when it is referred to
co-ordinate axes whose relation to it is arbitrary, it is, when
represented by the equation pq+m=0, determined by one con-
stant, and two straight lines that bear a fixed relation to it.
254. In the second form to which we have reduced the
general equation of the second order, viz. p?+ mq = 0, the curve
represented is a parabola, and is determined by one constant,
and two straight lines, one of which is arbitrary since one
constant more than necessary enters into the equation. It is
evident that p =0 is the equation to a diameter intersecting
q =0 at a point in the curve; and that g = 0 is the equation
to the tangent at that point, as it leads to p?=0, shewing that
its two points of intersection with the curve coincide with one
another.
255. When we take for the general equation of the
second degree the form
pat+mr = 0,
which contains seven constants, and combine it with either of
209
the equations p=0 or q=0, we get r°= 0; so that the two
straight lines represented by p=0, q=0, are tangents, and
the two points of contact lie in the straight line r=0; and
the curve is in this case determined by any two tangents and
the chord joining the points of contact.
Another form under which the general equation of the
/second degree may be written, is
hed “fs Av" ee [Ls
Uw, Vv, and r being, as before, linear functions of wv and y, each
involving two constants; and in this case each of the lines
u=0, v=0, 7 =0, represents the chord joining the points of
‘contact of the pair of real or imaginary tangents passing
through the intersection of the other two lines.
256. The general equation of the third degree
Yr Ayot+ Bye’ +Ca+Dy+ Eyet Fa?+ Gyt+He+I=0 (1),
provided 3° + 4z?+ Bs +C=0has no equal roots, can always
be transformed into
(ytaxr+b)(yt+a a+b’) (y+a"v+b") +m(y+ex+d)=0 (2);
only when the auxiliary cubic has imaginary roots, two
factors of the transformed equation are likewise imaginary,
but their product real.
| As equation (2) is of the third degree, and contains
the requisite number of independent constants, we ma
evidently assume it to be identical with (1); and upon
expanding and equating coefficients we find
@+a+a"'=A, aa +aa’+a'a" =B, ada’ = G.
to that — a, -—a’,-— a’, are the roots of the cubic equation
s+ A’+ Be+C=0 (3),
which we will suppose to be unequal.
_ For determining 3, b’, b”, we get three equations, which
aay be written
(b+ 0°40" =D, ab" +a" + a(b 4b") + (v7 +a’ )b=E,
: a(a'b" +a"0) +a'a"b = F (4).
14
Hi
HI
i
210
Now if a, a’, a’, be real and unequal, from these equa-
tions since they are linear, a single system of determined |
values of b, 0’, b, can be at once obtained. ‘But if two
roots of (3) a, a’, be imaginary, then a’ 4a’; and a a :
are real; and therefore from the same equations we can get)
one real system of determined values of :
|
ab a bb ro ee saris
|
consequently 6’ and 6” must be Sh Ls imaginary roots
of a quadratic equation since a’, a", are so; therefore bb” |
is real; as is also the product
|
(yt+dat+b)(y+a'a+b"’).
The three remaining constants m, c, and d, are given)
by the equations |
b(b' +b") +. 0b" +m=G, ab’b" + b(a'b’ + a’b)+mec= H
by'b" +md=T, (5); :
|
which, subject only to the condition of a, a’, a’, being
unequal, give one real system of determinate values of m,
c, d._ Hence it is proved that the proposed transformation
can be effected, and only in one way; none of the constants
of the Pehetariied equation being indeterminate, and none
of them having more than one hs |
957. If a’=a", equations (4) become |
b+(b'+b")=D, (at+a’) (b+ b")+20b=E, aa’ (b'+b")+a°b=F,
and cannot coexist unless the equation
Da? - Ea +F=0
is satisfied, in which case they will determine only 6, and
b'+6”; so that the transformation into (2) may be effected
in an infinite number of ways, as one of the quantities b’, b’,
may be assumed at pleasure; and the remaining constants
become known from equations (5). When the equation
Da? —-Ea+F=0 is not satisfied, it is impossible to put
211
the proposed equation into the form (2); and in place of it
we may choose the form
(yt+ax+b) {(y+a'x+b')?41 (y+ax+b)t+m (yt+ew+d) =0,
containing eight constants to which the number of constants
in the proposed is reduced on account of the condition a’= a’;
and by comparing coefficients it will be found that this trans-
formation can be effected in one determined way and no more.
When a= a' =a", equations (4) cannot coexist unless
E=2Da, F = Da’, in which éase they will determine only
the value of b+ 6'+ 5”. Consequently the transformation into
(2) may be effected in an infinite number of ways, as two of
the quantities b, b’, b” may be assumed at pleasure. Under
these conditions, the proposed equation has its independent con-
stants reduced to five; and besides the form (2) may likewise
be made to assume a form involving that number of constants,
viz.
(y+ ax+b)>+m(y+ca+d) =0.
When the conditions E = 2Da, F=Da’, are not satisfied,
it is impossible to put the proposed into the form (2); and
instead of it we may choose the form
(y+axv+ bi +lyt+ge+h)?+m(y+cr+d) <0,
containing one more constant than necessary, as the number
in the original equation is reduced to seven.
If we take the form
(y+au+ bP +i(y + ax +5) (yt+gu+h)+m=o,
containing only six constants, there will be an equation of
condition which we find to be Da?— Ea + F'=0; subject to
which the transformation can be effected in one determined
way and no more.
258, p);
but this last expression will be a maximum when p=2n-3-—p
or 2p =2n —3; and as p must be an integer, p=n — 1 or
p=n—2; and consequently we cannot have x greater than
1
@ (m — 1) (n — 2).
265. A curve of the m‘ order may in general have
m(m—1) tangents drawn to it through a given fixed point,
or parallel to a given line.
Let f(a, y) =0 be the equation to the curve, and
y=ma«-+ec the equation to a straight line; then if m and e
be such that f(«, m# +c) =0 has a real root twice repeated,
that root is the abscissa to the point of contact of the line with
the curve; and (Theory of Equations, Art. 60) is also a root
of f' (x, mx +c) =0; and if between these equations, which
are respectively of and of m—1 dimensions in & and Cc, we
eliminate c, there will result an equation of 2 (m —1) dimen-
sions, whose roots are the abscisse of the points of contact of
all tangents that can be applied to the curve parallel to
Y=mMmx; consequently the number of such tangents will in
general be n(n — 1).
But if the tangents are all to pass through a point (h, &),
then we must eliminate m between
fia, [m(w-h) +k]i =o, f’ Sa, [m (@ —h) + eae
and the result will have for its roots the abscisse of the points
of contact, in number 2(m—-—1) as before. When the given
point is in the curve, the tangent at that point must evidently
be reckoned twice ; and if it be in one of the asymptotes at an
infinite distance, the number of tangents that can pass through
it, or in other words that can be drawn parallel to the asym p-
tote, must be reduced by two units. Hence four tangents can
be drawn to a curve of the third order, parallel to one of its
asymptotes. If the given direction be parallel to the tangent
at a point of inflexion, or the given point be itself a point of
inflexion, the number of ‘tangents that can be drawn, will be
respectively diminished by one and two units.
266. When tangents are applied to a curve of the third
order respectively parallel to its three asymptotes, the points
of contact lie in a straight line.
As we may assume for the general equation to curves of
the third order any equation of the third degree in # and y
with nine independent constants, we may take for it the form
pqr+ms =0 (1);
but we cannot, as in the fundamental form where only the
simple power of s enters, be certain that this transformation
can be effected only in one way. The lines expressed by the
linear functions p, q, 7, s, now bear new relations to the curve ;
for if with equation (1) we combine the equation p= 0, we
get s’=0; therefore the line p=0 meets the curve in two
points only, and those points are coincident ; consequently, its
other point of intersection is at an infinite distance ; so that the
line »=0 is parallel to an asymptote, and also touches the
curve in the point where it intersects the line s = 0. Similarly,
gq = 0 and r = 0 represent lines parallel to the other two asymp;
totes, and touching the curve in the points where they inter-
sect s=0. Now four different tangents may be applied to the
curve parallel to any one of its asymptotes, and the four points
of contact may be joined by straight lines with the points of
contact of four tangents parallel to a second asymptote by six-
teen different straight lines, each of which will pass through
the point of contact of a tangent parallel to the third asymptote.
Since, therefore, there can be sixteen different systems of tan-
gents parallel to the asymptotes with their points of contact in:
a straight line, the general equation of the 3rd degree must be
capable of being put into the form (1) in the same number of
ways.
267. A-curve of the m™ order has in general 3n (n — 2)
points of inflexion.
Let f (a, y) = 0 be the equation to the curve, and
Y=MeL+C
the equation to a straight line; then if m and ¢ be such that,
219
J (v, ma +e) =0 has a real root thrice repeated, that root is
the abscissa to a point of inflexion; and it is (Theory of Equa-
tions, Art. 60) also a root of the equations
f'(#, mv+c)=0, f"(«v, me +c) =0;3
and if between these three equations m and c¢ be eliminated,
there will result an equation whose roots are the abscisse of
points of inflexion.
Now restoring the value of y, the three equations between
which the elimination is to be performed may be written
UW, =.0,
Un, + MV,_1 = 0,
Uy» + 2MW,_5 +m v,_. = 0,
where w,, 2,_,, &c. denote functions of m,n —1, &c. dimen-
sions in w and y; eliminating m between the two latter, we get
an equation of 3m — 4 dimensions, viz.
2 ra) ve
2V a—1 4Wy2Un 1 Un 1 0 Vn—9U a Os
Mm aw
u
and again eliminating y between this and uw, = 0, we finally get
an equation of » ($m —4) dimensions in w Now the curve will
have 7 rectilinear asymptotes, each of which will intersect it in
two points infinitely distant, which points have the character
of points of inflexion in that the radius of curvature at each
is infinitely great; therefore 22 points are included in the
above, which are not proper points of inflexion; and subtract-
ing, we get 37 (n — 2) for the number of points of inflexion
that a curve of the n™ order may in general have.
268. The points of inflexion of a curve of the third
order lie three and three in a straight line.
By the foregoing article a curve of the third order has in
general nine points of inflexion; and we shall now shew that
any straight line passing through two of them, must cut the
curve again in a third. As before, we may assume for the
general equation to curves of the third order, the form
pqr+ms’ =0,
since it contains the requisite number of independent constants.
If with this equation we combine the equation p = 0, we get
s’= 0; which shews that the three points in which the line
20
p =0 cuts the curve, are coincident; consequently the line
p =0 has acontact of the second order with the curve at the
point in which it intersects the line s = 0, and that point is a
point of inflexion. Similarly, each of the lines g=0, r=0
has a contact of the second order with the curve at the point
in which it intersects s = 0; and we thus obtain two other
points of inflexion, lying in the same straight line with the
first. Since out of nine things taken three at a time, twelve
combinations may be formed such that no two of them have
more than one element in common, it follows that the general
equation of the 3rd degree may in twelve ways be put into
the form pqr +ms*=0; but in only one of them will the
linear functions p, q, &c. be real, as only three of the points of
inflexion can be real and six of them imaginary.
269. When a curve of the third order with three real
asymptotes, has a double point, it will fall without, upon,
or within the ellipse which touches in their middle points the
three sides of the triangle formed by the asymptotes, according
as it is a proper double point, a cusp, or a conjugate point.
If wu = 0 be the equation to a curve, then
dint + dyyu. day =05
e e 0 e
and since at a double point d,y = ae at such a point we must
have du = 0, du = 0 (1); and to get the two values of d,y
at the double point, we must have recourse to the second derived
equation which, in consequence of the conditions (1), becomes
2 = g .
din + 2d.) Ay. doy + di, ut . (dy) = 0; (2);
and according as this equation gives real, equal, or imaginary
values of d,y, 1. e. according as
2 2 2
there will be a proper double point, a cusp, or a conjugate
point.
e 7 >
Now, taking the co-ordinate axes parallel to two asymp-~
totes, let
(nw +a)(y+b)\(e-na—y)+m(y+gxrt+h) =0,
or w= py (B—p—_q) +ms =0 be the equation to a curve of
the third order with three real asymptotes, where we have
puta+b+ce= 3; then
di. u =—2n'q, du =-— 2p, din, dU =n(r — p —q)3
therefore the condition (3) becomes
(r— p= 4g) — 4p 9 =p + ge Fr — 2 BS =. one 0.
which shews that a proper double point must fall without, and
a conjugate point within, that ellipse which is the locus of the
possible cusps; and which (Art. 263) touches in their middle
points the three sides of the triangle formed by the asymptotes.
270. But if a series of curves of the third order having
the same asymptotes, have each a double point such that one
of the tangents passing through it is in a constant direction,
then d,y will have a constant value « suppose, which must
satisfy equation (2). Hence substituting for d,y, and for
the differential coefficients of w their respective values, we get
for the locus of the double points the equation
rg+K«(2p+2q—B)+p=0
representing that diameter to whose chords the above-men-
tioned tangent is always parallel.
271. If a curve of the fourth order have three proper
double points, the six tangents at those points all touch a conic
section.
Since the curve has three double points, if p=0, g =0,
7 = 0, be the equations to three straight lines passing through
every two of them, and in the equation to the curve we
suppose p to vanish, qg’7* must appear as a factor; simi-
larly when g and vr are supposed to vanish, pr? and p 9
must appear respectively as factors; therefore the equation,
since it is of the fourth degree, must be of the form
epg + bp’r*+ ag’ — 2pqr (p+ q+er)=0 (1).
Now q = Xr represents any straight line passing through
the double point g=0, r=0; and if we combine it with
the equation to the curve and reject the factor * we obtain
for result
rar’ — 2pr (bd? + cr) + p? (ch? —2a’d +b) =0;
SS
Sa
7 ——— + 3ba*+2cx+d=03; this
latter equation must consequently be of the form
(2° + fu + g) (4a@ + v) = 0.
Hence we have the identical equation
(aa* + ba? + cu + dx +e) (4ax + v)
= (4aa° + 3ba* + 2cu + d) (av + tx + u);
and equating coefficients we get
a(v—4f+b)=0, bv -— 38bt-—4au+2ac=0,
cv — 2ct — 3bu + 3ad = 0,
dv—-dt—2cu+4ae=0, ev=du;
substituting the values of ¢ and w given by the first and last
of these equations in the other three, we get three values of
od, which equated two and two give
225
(4d — 16ae) (Gad — be) = (8ac — 3b’) (cd - Obe),
(bd — 16ae) (6be — cd) = (8ce — 3d’) (be — Gad);
and these two equations contain only the unknown quantities
m and n, and serve to determine them.
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