S. L. (Sidney Luxton) Loney.

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their centres is the curve

{x^ + y"^) [a^x^ + b^y^ + a%^) = <?■ {o?x^ + b^y'^) .

20. The polar of a point P with respect to an ellipse touches a
fixed circle, whose centre is on the major axis and which passes
through the centre of the ellipse. Shew that the locus of P is a
parabola, whose latus rectum is a third proportional to the diameter
of the circle and the latus rectum of the ellipse.

21. Prove that the locus of the pole, with respect to the ellipse, of


^ v^ 1
any tangent to the auxiliary circle is the curve -4 + n = -^ •


22. Shew that the locus of the pole, with respect to the auxiliary
circle, of a tangent to the ellipse is a similar concentric ellipse,
whose major axis is at right angles to that of the original ellipse.

23. Chords of the ellipse touch the parabola ay^= -2b^x; prove
that the locus of their poles is the parabola ay^ = 2b^x.

24. Prove that the sum of the angles that the four normals
drawn from any point to an ellipse make with the axis is equal to
the sum of the angles that the two tangents from the same point
make with the axis.

[Use the equation of Art. 268.]

25. Triangles are formed by pairs of tangents drawn from any
point on the ellipse

2 2

a2a;2 + hY = {a^ + ^^f to the ellipse -g + p = 1,

and their chord of contact. Prove that the orthocentre of each such
triangle lies on the ellipse.

26. An ellipse is rotated through a right angle in its own plane
about its centre, which is fixed ; prove that the locus of the point of
intersection of a tangent to the ellipse in its original position with
the tangent at the same point of the curve in its new position is

(a;2 + 2/2) (a;2 + 1,2 _ ^2 _ ^2) ^ 2 {a^ - b^) xy.

27. If y and Z be the feet of the perpendiculars from the foci
upon the tangent at any point P of an ellipse, prove that the tangents
at Y and Z to the auxiliary circle meet on the ordinate of P and that
the locus of their point of intersection is another ellipse.

28. Prove that the directrices of the two parabolas that can be
drawn to have their foci at any given point P of the ellipse and to
pass through its foci meet at an angle which is equal to twice the
eccentric angle of P.

29. Chords at right angles are drawn through any point P of the

ellipse, and the line joining their extremities meets the normal in the

point Q. Prove that Q is the same for all such chords, its

,. , , . a^e2cosa , -a2&e2sina
coordinates being — ^ — 717- and 5 — z^ — .

Prove also that the major axis is the bisector of the angle PGQ,
and that the locus of Q for different positions of P is the ellipse

a;2 y^ _

\^^+by '



295. The hyperbola is a Conic Section in which the
eccentricity e is greater than unity.

To find the equation to a hyperbola.

Let ZK be the directrix, 8 the focus, and let SZ be
perpendicular to the directrix.

There will be a point A on AZ^ such that

SA - ^e.AZ (ly


Since e> 1, there will be another point A\ on /S'^ pro-
duced, such that

SA! = e.A'Z. (2).

Let the length A A! be called 2c&, and let C be the middle
point of AA! .

Subtracting (1) from (2), we have

= e\CA' + GZ'\-e\_GA - GZ] = e,2GZ,

i.e. GZ=- , (3).

Adding (1) and (2), we have

e (AZ +A'Z) = SA' + SA = 2GS,

i,e. e.AA' = 2.GS,

and hence GS = ae (4).

Let G be the origin, GSX the axis of x, and a straight
line GT, through G perpendicular to GX, the axis of y.

Let P be any point on the curve, whose coordinates are
X and y, and let FM be the perpendicular upon the directrix,
and PiV the perpendicular on ^^^.

The focus aS' is the point (ae, 0).

The relation JSP"" = e^ . PM^ = e^ . ZN^ then gives

{x — aef + y^ = e^\ x

i.e. x^ — 2aex + a^e^ + 2/^ = &^x^ — ^aex + a^.

Hence x" {e" - I) - y'^ = or {e" - \\

x^ v
i.e. - -— ^— - ==1 (5).

Since, in the case of the hyperbola, e> 1, the quantity
a^ (e^ — 1) is positive. Let it be called b% so that the equa-
tion (5) becomes

X2 y2

r2-b2=^ (^)'

where b'^=^a'e^-a^^GjS^-GA' (7),

and therefore GS^ = a^ + b^ „ . . . (8).



296. The equation (6) may be written
y^ s^ ^ _x- - a^ _{x — a) {x + a)


so that PN^ : AJV . N"!' :: b' : d\

If we put x^O in equation (6), we have y'^- — lf,
shewing that the curve meets the axis CZ in imaginary

Def. The points A and A' are called the vertices of the
hyperbola, C is the centre, -4^' is the transverse axis of the
curve, whilst the line BB' is called the conjugate axis,
where B and B' are two points on the axis of y equidistant
from C, as in the figure of Art. 315, and such that

B'G = CB^h.

297. Since S is the point (ae, 0), the equation referred to the
focus as origin is, by Art. 128,

{x + ae)^ 2/2 _

~"^2 p-1'

i.e. ^'+2 - |' + e2-l = 0.

Similarly, the equations, referred to the vertex A and foot of the
directrix Z respectively as origins, wUl be found to be

x^ y^ 2x ^

, a;2 w2 2a; ^ 1

and 1^ +__=!_

a^ b'^ ae e^

^ The equation to the hyperbola, whose focus, directrix, and eccen-
tricity are any given quantities, may be written down as in the case
of the ellipse (Art. 249).

298. There exist a second focus and a second directrix
to the curve.

On SO produced take a point S', such that
and another point Z', such that


ZC = CZ' =



Draw Z'M' perpendicular to AA^ and let FM be pro-
duced to meet it in M' .

The equation (5) of Art. 295 may be written in the

tc^ + ^aex + cC-G^ + 2/^ = ^^ + ^aex + a^,

i.e. {x + aef + 2/^ = e^(-+a3j ,

i.e. S'P^ = e" {Z'G + CNf = e^ . PM\

Hence any point F of the curve is such that its distance
from S' is e times its distance from Z'K\ so that we should
have obtained the same curve if we had started with S' as
focus, Z'K^ as directrix, and the same eccentricity e.

299. The difference of the focal distances of any point
on the hyperbola is equal to the transverse axis.

For (Fig., Art 295) we have

SP = e.PM, and 8'P = e.PM'.

Hence S'P - SP = e{PM' - PM) = e . MM'

= e.ZZ' = 2e.CZ=2a

— the transverse axis AA'.

Also SP = e.PM^e. ZN ^ e.CN-e. CZ^ ex' - a,

and 8'P^e. PM' = e . Z'N =e . C]^+ e .Z'C^bk' + a.,

where x' is the abscissa of the point P referred to the centre
as origin.

300. Latus-rectum of the Hyperbola.

Let LSL' be the latus-rectum, i.e. the double ordinate
of the curve drawn through S.

By the definition of the curve, the semi-latus-rectum SL
= e times the distance of L from the directrix


= e . CS — eCZ=ae^ ~a = — ,


by equations (3), (4), and (7) of Art. 295.


301. To trace the curve

- ^=1 (1)

The equation may be written in either of the forms

^=**\/5^ (^)'

or a! = ±fli /^ + 1 (3).


From (2), it follows that, if ^ < cf?^ i.e. if x lie between a
and — (X, then y is impossible. There is therefore no part
of the curve between A and A! .

For all values of oi? > o?- the equation (2) shews that
there are two equal and opposite values of y^ so that the
curve is symmetrical with respect to the axis of x. Also,
as the value of x increases, the corresponding values of y
increase, until, corresponding to an infinite value of x^ we
have an infinite value of y.

For all values of y^ the equation (3) gives two equal
and opposite values to cc, so that the curve is symmetrical
with respect to the axis of y.

If a number of values in succession be given to x^ and
the corresponding values of y be determined, we shall
obtain a series of points, which will all be found to lie on a
curve of the shape given in the figure of Art. 295.

The curve consists of two portions, one of which extends
in an infinite direction towards the positive direction of
the axis of £c, and the other in an infinite direction towards
the neofative end of this axis.


302. The quantity — 2 — w — 1 '^^ positive, zero^ or

negative, according as the point {x, y') lies within, u2Jon,
or without, the curve.

Let Q be the point {x, y'), and let the ordinate QW



through Q meet the curve in P, so that, by equation (6) of
Art. 295,

^ _ FIP_ _

and hence —to~ = — ^ — 1-

If Q be within the curve then y, i.e. QN, is less than
Pi\^, sothat |^<_^-, ^.e.<__l.

Hence, in this case, — -j->0, i.e. is positive.



Similarly, if Q be without the curve, then y' > PN, and
we have -^ - ^ — 1 negative.

303. To find the length of any central radius dravjn in
a given direction.

The equation (6) of Art. 295, when tran^rred to polar
coordinates, becomes

(cos- e _Biv? e\

1 COS- 6 sin^e cos-e/b' , . ,\ ,,,

i5=^ - ^^=-j-. - (;?-ten^^j (1).

This is the equation giving the value of any central
radius of the curve drawn at an inclination 6 to the trans-
verse axis.


So long as tan^ ^< -^, the equation (1) gives two equal

and opposite values of r corresponding to any value of 0.


For values of tan^ 0> —^, the corresponding values of


— are negative, and the corresponding values of r imaginary.
Any radius drawn at a greater inclination than tan~^ -


does not therefore meet the curve in any real points, so
that all the curve is included within two straight lines

drawn through C and inclined at an angle ± tan~-^ — to CX.

Writing (1) in the form

7* -—


we see that r is least when the denominator is greatest, i.e.
when ^ = 0. The radius vector CA is therefore the least.

Also, when tan ^ = ± - , the value of r is infinite.

For values of between and tan"^ - the corresponding


positive values of r give the portion ^^ of the curve (Fig.,

Art. 295) and the corresponding negative values give the

portion A'R'.


For values of 6 between and — tan~^ - , the positive


values of R give the portion AR^, and the negative values

give the portion A'R^.

The ellipse and the hyperbola since they both have a
centre (7, such that all chords of the conic passing through
it are bisected at it, are together called Central Conics.

304. In the hyperbola any ordinate of the curve does
not meet the circle on A A' as diameter in real points.
There is therefore no real eccentric angle as in the case of
the ellipse.

"When it is desirable to express the coordinates of any
point of the curve in terms of one variable, the substitutions

X = a sec ^ and y = b tan <f>

may be used; for these substitutions clearly satisfy the
equation (6) of Art. 295.

The angle ^ can be easily defined geometrically.

On AA! describe the auxiliary circle, (Fig., Art. 306)


and from the foot iV of any ordinate NP of the curve draw
a tangent NU to this circle, and join CU. Then


i.e. x = GN=:a^QGNCU.

The angle NCU is therefore the angle <^.

Also HU = CU tan (j> = a ta>ii^,

so that JV^F : JSFU :: b : a.

The ordinate of the hyperbola is therefore in a constant
ratio to the length of the tangent drawn from its foot to
the auxiliary circle.

This angle eft is not so important an angle for the
hyperbola as the eccentric angle is for the ellipse.

305. Since the fundamental equation to the hyper-
bola only differs from that to the ellipse in having — b^
instead of b% it will be found that many propositions for
the hyperbola are derived from those for the ellipse by
changing the sign of b^.

Thus, as in Art. 260, the straight line y — mx + c meets
the hyperbola in points which are real, coincident, or
imaginary, according as

c^> — < a^Tn^ — b^.
As in Art. 262, the equation to the tangent at (x, y'^ is

As in Art. 263, the straight line

y = mx + sJa^TYi^ — b^
is always a tangent.
The straight line

X cos a + y sin a = j^
is a tangent, if p^ = a^ cos^ a — b^ sin^ a.

The straight line Ix + my = n
is a tangent, if ?i^ — aH^ - ¥m^. [Art. 264.]


The normal at the point (x\ y') is, as in Art. 266,


306. With some modifications the properties of Arts.
269 and 270 are true for the hyperbola also, if the
corresponding figure be drawn.

In the case of the hyperbola the tangent bisects the
interior, and the normal the exterior, angle between the
focal distances SP and S'P.

It follows that, if an ellipse and a hyperbola have the
same foci aS' and S', they cut at right angles at any common
point P. For the tangents in the two cases are respec-
tively the internal and external bisectors of the angle SPS',
and are therefore at right angles.

307. The equation to the straight lines joining the
points {a sec ^, h tan (j>) and {a sec <^\ h tan ^') can be
shewn to be

X 4>' — ^ y . ct> + c{i 4> + <h'

- cos ^ — 7- sm ^ = cos ^ .
a 2 2 2


Hence, by putting (f>' — <^, it follows that the tangent at
the point {a sec ^, h tan ^) is

1- sin </) = cos d).

a 6

It could easily be shewn that the equation to the
normal is

ax sin <^ + by =^{a^ + b^) tan ^.

308. The proposition of Art. 272 is true also for the

As in Art. 273, the chord of contact of tangents
from (iTj, 2/i) is

^^1 _ y^j _ -1

As in Art. 274, the polar of any point (x^, 2/1) is

^^ y^i __ 1

a^ 52 - •

As in Arts. 279 and 281, the locus of the middle
points of chords, which are parallel to the diameter y = 7nx,
is the diameter y = m-^x, where

The proposition of Art. 278 is true for the hyperbola
also, if we replace b^ by — b\

309. Director circle. The locus of the intersection
of tangents which are at right angles is, as in Art. 271,
found to be the circle x^ + y'^ = a^ — b% i. e. a circle whose
centre is the origin and whose radius is Ja^ — b\

If 5^ < a^, this circle is real.

If b^ — a^, the radius of the circle is zero, and it reduces
to a point circle at the origin. In this case the centre is
the only point from which tangents at right angles can be
drawn to the curve.

If y^ > a^, the radius of the circle is imaginary, so that
there is no such circle, and so no tangents at right angles
can be drawn to the curve.


310. Equilateral^ or Rectangular^ Hyperbola.

The particular kind of hyperbola in which the lengths
o£ the transverse and conjugate axes are equal is called an
equilateral, or rectangular, hyperbola. The reason for the
name "rectangular" will be seen in Art. 318.

Since, in this case, b = a, the equation to the equilateral
hyperbola, referred to its centre and axes, is x^ -y^ = a\

The eccentricity of the rectangular hyperbola is ^2.

For, by Art. 295, we have, in this case,
, a'' + ¥ 2t*2
a^ a^

so that e — J2.

311. Ex. The perpendiculars from the centre upon the tangent
and normal at any point of the hyperbola -^—j-^ = l meet them in Q

and R. Find the loci of Q and R.

As in Art. 308, the straight line

X cos a + y sin a =p

is a tangent, if p^ — a^ cos^ a-h^ sin^ a.

But p and a are the polar coordinates of Q, the foot of the perpen-
dicular on this straight line from C.

The polar equation to the locus of M is therefore

r2 = a2cos2^-62sin2 6',

i.e., in Cartesian coordinates,

(x^ + y^f^a^x^-h^y^

If the hyperbola be rectangular, we have a = &, and the polar
equation is

r2 = a2 (cos2 d - sin2 6) = a^ cos 26.
Again, by Art. 307, any normal is

axsin(t) + hy = {a^ + h^)ia,n<() (1).

The equation to the perpendicular on it from the origin is

hx-ayB\n = (2).

If we eliminate 0, we shall have the locus of R.

From (2), we have sind>= — ,


sin hx

and then tan = / :=r — — j- — .

>/ 1 - sin^ ^Ja^y^— h^x^

Substituting in (1) the locus is

{x^ + 2/2)2 i^aY - b^x^) = (a2 + ^2)2 ^y.



Find the equation to the hyperbola, referred to its axes as axes of

1. whose transverse and conjugate axes are respectively 3 and 4,

2. whose conjugate axis is 5 and the distance between whose foci
is 13^

3. whose transverse axis is 7 and which passes through the point
(3, -2),

4. the distance between whose foci is 16 and whose eccentricity


5. In the hyperbola 4a;2- 9i/^ = 36, find the axes, the coordinates
of the foci, the eccentricity, and the latus rectum.

6. Find the equation to the hyperbola of given transverse axis
whose vertex bisects the distance between the centre and the focus.

7. Find the equation to the hyperbola, whose eccentricity is |,
whose focus is {a, 0), and whose directrix is 4:X-dy = a.

Find also the coordinates of the centre and the equation to the
other directrix.

8. Find the points common to the hyperbola 25a;2- 9^/^=225
and the straight line 25a; + 12?/ -45 = 0.

9. Find the equation of the tangent to the hyperbola 4x'^ - 9y^=l
which is parallel to the line 4oy = 5x + l.

10. Prove that a circle can be drawn through the foci of a
hyperbola and the points in which any tangent meets the tangents at
the vertices.

11. An ellipse and a hyperbola have the same principal axes.
Shew that the polar of any point on either curve with respect to the
other touches the first curve.

12. In both an ellipse and a hyperbola, prove that the focal
distance of any point and the perpendicular from the centre upon the
tangent at it meet on a circle whose centre is the focus and whose
radius is the semi-transverse axis.

CC U 5/ '?/ 1

13 Prove that the straight lines — j- = m and - + ^ = - always
^^' ° ah a m

meet on the hyperbola.

14. Find the equation to, and the length of, the common tangent
to the two hyperbolas -^ - p = l and —^- t^=1-

15. In the hyperbola 16^2-9^2 -144^ find the equation to the
diameter which is conjugate to the diameter whose equation is x=2,y.


16. Find the equation to the chord of the hyperbola

which is bisected at the point (5, 3).

17. In a rectangular hyperbola, prove that


18. the distance of any point from the centre varies inversely as
the perpendicular from the centre upon its polar.

19. if the normal at P meet the axes in G and g, then PG=Pg=PC.

20. *lie angle subtended by any chord at the centre is the
supplement of the angle between the tangents at the ends of the

21. the angles subtended at its vertices by any chord which is
parallel to its conjugate axis are supplementary.

22. The normal to the hyperbola —, ~ ^ = 1 meets the axes in M

and N, and perpendiculars MP and NP are drawn to the axes ; prove
that the locus of P is the hyperbola

23. If oiie axis of a varying central conic be fixed in magnitude
and position, prove that the locus of the point of contact of a tangent
drawn to it from a fixed point on the other axis is a parabola.

24. If the ordinate MP of a hyperbola be produced to Q, so that
MQ is equal to either of the focal distances of P, prove that the locus
of Q is one or other of a pair of parallel straight lines.

25. Shew that the locus of the centre of a circle which touches
externally two given circles is a hyperbola.

26. On a level plain the crack of the rifle and the thud of the ball
striking the target are heard at the same instant; prove that the
locus of the hearer is a hyperbola.

27. Given the base of a triangle and the ratio of the tangents of
half the base angles, prove that the vertex moves on a hyperbola
whose foci are the extremities of the base.

28. Prove that the locus of the poles of normal chords with

respect to the hyperbola — , - '^ = 1 is the curve

a^ b^

y^a^ - x%^ = (a2 + 62)2 3.2^2,

29. Find the locus of the pole of a chord of the hyperbola which
subtends a right angle at (1) the centre, (2) the vertex, and (3) the
focus of the curve.

30. Shew that the locus of poles with respect to the parabola
y^=^ax of tangents to the hyperbola x^-y^=a^ is the ellipse
4a;2 + 2/2=4a2.


31. Prove that the locus of the pole with respect to the hyperbola

—„ = \ of any tangent to the circle, whose diameter is the line

a?- W-

3j 11 1

ng the foci, is the ellipse — 4 + ri =

a^ ¥ a^+b^

32. Prove that the locus of the intersection of tangents to a
hyperbola, which meet at a constant angle /3, is the curve

33. From points on the circle x^ + y^=a^ tangents are drawn to
the hyperbola x^ - y'^=a^; prove that -the locus of the middle points of
the chords of contact is the curve

(a;2 - 1/2)2 _ ^2 ^^2 ^ y2j_

34. Chords of a hyperbola are drawn, all passing through the
fixed point {h, 1c) ; prove that the locus of their middle points is a

hyperbola whose centre is the point ( ^ , - J , and which is similar to

either the hyperbola or its conjugate.

312. Asymptote. Def. An asymptote is a straight
line, which meets the conic in two points both of which are
situated at an infinite distance, but which is itself not alto-
gether at infinity.

313. To find the asymptotes of the hyperbola

As in Art. 260, the straight line

y = 7nx + c (1)

meets the hyperbola in points, whose abscissae are given by
the equation

x" {¥ - a'Tiv')- 2a^mcx - a^ (c^ + 6^) = Q (2).

If the straight line (1) be an asymptote, both roots of (2)
must be infinite.

Hence (C. Smith's Algebra, Art. 123), the coefiicients of
x^ and X in it must both be zero.

We therefcwre have

h^ — a^jn^ = 0, and a^mc — 0.


Hence m = =*=-, and c = 0.


Substituting these values in (1), we have, as the^fc
quired equation, ^^


-W =: =fc - aj.

^ a

There are therefore two asymptotes both passing
through the centre and equally inclined to the axis of x,
the inclination being

tan"-^ — .

The equation to the asymptotes, written as one equa-
tion, is

Cor. For all values of c one root of equation (2) is

infinite if 7?2 = ± — . Hence any straight line, which is

parallel to an asymptote, meets the curve in one point at
infinity and in one finite point.

314. That the asymptote passes through two coincident points
at infinity, i. e. touches the curve at infinity, may be seen by finding
the equations to the tangents to the curve which pass through any

point f rcj , — ajj^ j on the asymptote y=-x.

As in Art. 305 the equation to either tangent through this point is
y = mx + Ja^m^ - b^^

where - ar, = mx, -{• Ja^m^ - 6^

a ^ ^

i.e. on -clearing of surds,

m2 {x^^ - a2) - 2m - x-,^+ (x.^ + a^) -,=0.

One root of this equation is m = -, so that one tangent through
the given point \^y=:- x, i.e. the asymptote itself.



315. Geometrical construction fo7' the asymptotes.

Let A' A be the transverse axis, and along the conju-
gate axis measure off CB and CB\ each equal to h.
Through B and B' draw parallels to the transverse axis
and through A and A' parallels to the conjugate axis, and
let these meet respectively in K^, K^^ K^, and K^, as in the

Clearly the equations of K^CK^ and K^CK^ are

h , b

y = — x, and y = x,

^ a ' ^ a '

and these are therefore the equations of the asymptotes.

316. Let any double ordinate FlSfP' of the hyperbola
be produced both ways to meet the asymptotes in Q and Q\
and let the abscissa CN be x.

Since P lies on the curve, we have, by Art. 302,



Since Q is on the asymptote whose equation is 3/ = - x,

we have NQ = - £c


Hence PQ = NQ - NP = ^- (x' - slx'^ - a\

and QP' ^^- (x' + sj^-a\

a ^ '

Therefore PQ .QP' = ^ jcc'^ - (x'^ - a')} = h\

Hence, if from any point on an asymptote a straight
line be drawn perpendicular to the transverse axis, the
product of the segments of this line, intercepted between
the point and the curve, is always equal to the square on
the semi-conjugate axis.


PQj-{^-^W^r^) = ^

« ax' + Jx'''-a''


x' + \/x^ — a^

PQ is therefore always positive, and therefore the
part of the curve, for which the coordinates are positive,
•is altogether between the asymptote and the transverse

Also as X increases, i. e. as the point P is taken further
and further from the centre C, it is clear that PQ con-
tinually decreases ; finally, when x' is infinitely great, PQ
is infinitely small.

The curve therefore continually approaches the asymp-
tote but never actually reaches it, although, at a very great
distance, the curve would not be distinguishable from the

This property is sometimes taken as the definition of an

317. If SF be the perpendicular from ^S' upon an
asymptote, the point F lies on the auxiliary circle. This

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