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288 COORDINATE GEOMETRY,

follows from the fact that the asymptote is a tangent,

whose point of contact happens to lie at infinity, or it may

be proved directly.

For

GF= CS cos FCS =CS.%= jÂ¥+Â¥ . -.-^^ - a.

Also Z being the foot of the directrix, we have

GA^= CS.cz, (Art. 295)

and hence CF^ = CS . CZ, i.e. CS : CF :: CF : CZ.

By Euc. YI. 6, it follows that l CZF= l CFS= a right

angle, and hence that F lies on the directrix.

Hence the perpendiculars from the foci on either asymptote

meet it in the sam,e points as the corresponding directrix,

and the common points of intersection lie on the auxiliary

circle.

318. Equilateral or Rectangular Hyperbola.

In this curve (Art. 310) the quantities a and b are equal.

The equations to the asymptotes are therefore y=.^x, i.e.

they are inclined at angles =t 45Â° to the axis of x, and hence

they are at right angles. Hence the hyperbola is generally

called a rectangular hyperbola.

319. Conjugate Hyperbola. The hyperbola which

has BB' as its transverse axis, and A A' as its conjugate

axis, is said to be the conjugate hyperbola of the hyperbola

whose transverse and conjugate axes are respectively AA'

and BB'.

Thus the hyperbola

Â¥ a'~^ ^ ^'

is conjugate to the hyperbola

^-^ = 1 (2).

a' b' ^ ^

Just as in Art. 313, the equation to the asymptotes of

F~a'

11^ ex?

X-

THE HYPERBOLA. CONJUGATE DIAMETERS. 289

which, by the same article, is the equation to the asymp-

totes of (2).

Thus a hyperbola and its conjugate have the sam.e

asymptotes.

The conjugate hyperbola is the dotted curve in the

figure of Art. 323.

320. Intersections of a hyperbola with a pair of con-

jugate diameters.

The straight line y = m^x intersects the hyperbola

a^ b'

in points whose abscissae are given by

i.e. by the equation a^ â€” â€” â€” - .

^ 0^ â€” a^m^

The points are therefore real or imaginary, according as

a^TYi^ is < or > 6^,

i.e. according as

m^ is numerically < or > - (1),

a

i.e. according as the inclination of the straight line to the

axis of X is less or greater than the inclination of the

asymptotes.

Now, by Art. 308, the straight lines y - tyi^x and y = m^x

are conjugate diameters if

^' /ox

Â»^1^2 = ^ (2).

Hence one of the quantities m^ and m^ must be less

than - and the other greater than - .

a a

Let mj be < -, so that, by (1), the straight line y = m^x

meets the hyperbola in real points.

L. 19

290 COOEDINATE GEOMETRY.

Then, by (2), m^ must be > - , so that, by (1), the straight

CL

line y = m^x will meet the hyperbola in imaginary points.

It follows therefore that only one of a pair of conjugate

diameters meets a hyperbola in real points.

321. If a pair of diameters he conjugate with respect

to a hyperbola, they will he conjugate with respect to its con-

jugate hyperhola.

Eor the straight lines y = tyi^ and y = m^ are conjugate

with respect to the hyperbola

a^ y"

a

-|.-1 (1).

if ^i'^2==-2 (2).

Cv

Now the equation to the conjugate hyperbola only

differs from (1) in having â€” a^ instead of a^ and - h^ instead

of h% so that the above pair of straight lines will be con-

jugate with respect to it, if

'^â„¢= = l-^ = a^ (^)-

But the relation (3) is the same as (2).

Hence the proposition.

322. If a pair of diameters he conjugate with respect

to a hyperhola, one of them meets the hyperhola in real points

and the other meets the conjugate hyperhola in real points.

Let the diameters be 3/ = m^x and y = m^x, so that

h^

m^m^ = â€” .

a^

As in Art. 320 let mi < - , and hence m- > - , so that the

^ a' ^ Â«'

straight line y â€” m^x meets the hyperbola in real points.

Also the straight line y â€” m^x meets the conjugate

hyperbola ^ -^-\ in points whose abscissae are given by

a

a^y

the equation x^ ( -zj A â€” \y i.e. by .t^ =

ra^ a^ â€” h"^'

THE HYPERBOLA. CONJUGATE DIAMETERS. 291

Since mo > - , these abscissae are real.

" a

Hence the proposition.

323. If aiKiir of conjugate diameters meet the hyperbola

and its conjugate in P and D, then (1) CP^ â€” CD^ = a^ â€” 6^,

and (2) the tangents at P, D and the other ends of the

diameters passing through them form a 'parallelogram whose

vertices lie on the asymptotes and whose area is constant.

Let P be any point on the hyperbola â€” â€” ^ = 1 whose

coordinates are (a sec cf), h tan ^).

The equation to the diameter CP is therefore

h tan 4> ^ â€¢ I

y = J x = x . - sm <p.

a sec <^

a

By Art. 308, the equation to the straight line, Avhich

is conjugate to CP, is

b

y ^x â€” . â€” -

a sm <fi

19â€”2

292 COORDINATE GEOMETRY.

This straight line meets the conjugate hyperbola

in the points (a tan <^, h sec ^), and (â€” a tan (j>, â€”h sec ^) so

that D is the point {a tan <^, 6 sec ^).

We therefore have

CP" - a' sec^ Â«^ + 6Han2 </,,

and CZ)^ - a^ tan^ <^ + Â¥ sec^ <^.

Hence

CP2 _CD^^{a''- h^) (sec2 ^ - tan^ </>) = c*2 - 51

Again, the tangents at P and Z> to the hyperbola and

the conjugate hyperbola are easily seen to be

f sind) =^cos d), (1).

a ^ \ /'

If Q^

and 7 sind) = cosd) (2).

a ^ '

These meet at the point

X y cos ^

a h 1 â€” sin Â«^ *

This point lies on the asymptote CL.

Similarly, the intersection of the tangents at P and D '

lies on CL-[^ that of tangents at D' and F' on GL' ^ and

those at D and F' on CZj.

If tangents be therefore drawn at the points where a

pair of conjugate diameters meet a hyperbola and its

conjugate, they form a parallelogram whose angular points

are on the asymptotes.

Again, the perpendicular from C on the straight line (1)

cos ^ ah cos

y

1 1 . , , V*" + a' sin' A

a6 Â«6 ah

sJhHec^<j> + d'tQ.n^4> CD FK'

THE CONJUGATE HYPERBOLA. 293

SO that PK X perpendicular from G on TK ~ ah,

i.e. area of the parallelogram CPKD â€” ah.

Also the areas of the parallelograms CPKD, CDK^P\

CP'K'D', and CB'K;P are all equal.

The area KK^K'K^ therefore = 4a6.

Cor. PK^ CD = D'C = K^P, so that the portion of a

tangent to a hyperbola intercepted between the asymptotes

is bisected at the point of contact.

324. Relation hefweert the equation to the hyperhola,

the equation to its asymptotes, and the equation to the conju-

gate hyperbola.

The equations to the hyperbola, the asymptotes, and the

conjugate hyperbola are respectively

5-F = i â€¢ W-

%-%-' -(2)-

and-^-|^ = -l (3).

a^ 0"

We notice that the equation (2) differs from equation (1)

by a constant, and that the equation (3) differs from (2) by

exactly the same quantity that (2) differs from (1).

If now we transform the equations in any way we

please â€” by changing the origin and directions of the axes â€”

by the most general substitutions of Art. 132 and by

multiplying the equations by any â€” the same â€” constant,

we shall alter the left-hand members of (1), (2), and (3) in

exactly the same way, and the right-hand constants in the

equations will still be constants, and differ in the same way

as before.

Hence, whatever be the form of the equation to a

hyperbola, the equation to the asymptotes only differs from

it by a constant, and the equation to the conjugate

hyperbola differs from that to the asymptotes by the same

constant.

294 COORDINATE GEOMETRY.

325. As an example of the foregoing article, let it be required

to find the asymptotes of the hyperbola

Sx^-5xy-2y^ + 5x+ny-8 = ...(1).

Since the equation to the asymptotes only differs from it by a

constant, it must be of the form

Sx^-5xy-2y^ + 5x + lly + c = (2).

Since (2) represents the asymptotes it must represent two straight

lines. The condition for this is (Art. 116)

3(-2)c + 2.|.-V-(-f)-3(W-(-2)(t)2-c(-t)2=0,

i.e. c=-12.

The equation to the asymptotes is therefore

3a;2 - 5xy - 2tf + 5a; + 11?/ - 12 = 0,

and the equation to the conjugate hyperbola is

3a;2 - 5xy -2i/ + 5x + lly -16 = 0.

326. As another example we see that the equation to any

hyperbola whose asymptotes are the straight lines

Ax + By + G = and A-^x + B-^y + 0-^ = 0,

is {Ax + By + C){A^x + B^y + Gj) = \'' (1),

where X is any constant.

For (1) only differs by a constant from the equation to the

asymptotes, which is

{Ax + By + C){A^x + B^y + G^) = (2).

If in (1) we substitute - A^ for X^ ^e shall have the equation to its

conjugate hyperbola.

It follows that any equation of the form

[Ax + By + C) {A^x + B-^y + Oj) = X^

represents a hyperbola whose asymptotes are

Ax + By + C = 0, and A^x + B^y + G^ = 0.

Thus the equation x{x + y) = a^ represents a hyperbola whose

asymptotes are x = and x + y = 0.

Again, the equation x'^ + 2xy cot 2a - y^ = a'^,

i.e. {x cot a- y) {x ta.n a + y) = a^,

represents a hyperbola whose asymptotes are

X cot a-y = 0, and x tan a + y = 0.

327. It would follow from the preceding articles that the

equation to any hyperbola whose asymptotes are x = and ?/ = is

a;2/ = const.

THE HYPERBOLA. EXAMPLES. 295

The constant could be easily determined in terms of the semi-

transverse and semi-conjugate axes.

In Art. 328 we shall obtain this equation by direct transformation

from the equation referred to the principal axes.

EXAMPLES. XXXVII.

1. Through the positive vertex of the hyperbola a tangent is

drawn; where does it meet the conjugate hyperbola?

2. lie and e' be the eccentricities of a hyperbola and its conjugate,

prove that _ + _=!.

3. Prove that chords of a hyperbola, which touch the conjugate

hyperbola, are bisected at the point of contact.

4. Shew that the chord, which joins the points in which a pair of

conjugate diameters meets the hyperbola and its conjugate, is parallel

to one asymptote and is bisected by the other.

5. Tangents are drawn to a hyperbola from any point on one of

the branches of the conjugate hyperbola; shew that their chord of

contact will touch the other branch of the conjugate hyperbola.

6. A straight line is drawn parallel to the conjugate axis of a

hyperbola to meet it and the conjugate hyperbola in the points P and

Q ; shew that the tangents at P and Q meet on the curve

6^'

4x^

and that the normals meet on the

axis

of a;.

7. From a point G on the transverse axis GL is drawn perpen-

dicular to the asymptote, and GP a normal to the curve at P. Prove

that LP is parallel to the conjugate axis.

8. Find the asymptotes of the curve 2x'^ + 5xy + 2ij^ -\-4:X + 5y = 0,

and find the general equation of all hyperbolas having the same

asymptotes.

9. Find the equation to the hyperbola, whose asymptotes are the

straight lines x + 2y + 3 = 0, and Sx + 4y + 5 = 0, and which passes

through the point (1,-1).

Write down also the equation to the conjugate hyperbola.

10. In a rectangular hyperbola, prove that CP and CD are equal,

and are inclined to the axis at angles which are complementary.

296

COORDINATE GEOMETRY. [ExS. XXXVII.]

11. C is the centre of the hyperbola -3 - t2 = 1 and the tangent at

any point P meets the asymptotes in the points Q and R. Prove that

the equation to the locus of the centre of the circle circumscribing

the triangle CQB is 4 {a^x^ - b^) = (a^ + b^f.

12. A series of hyperbolas is drawn having a common transverse

axis of length 2a. Prove that the locus of a point P on each hyper-

bola, such that its distance from the transverse axis is equal to its

distance from an asymptote, is the curve {x^-y^)'^=4:X^{x^-a^).

328. To jind the equation to a hyperbola referred to its

asymptotes.

Let P be any point on the hyperbola, whose equation

referred to its axes is

,2 '" ^

.(1).

a" 52

Draw PH parallel to one asymptote CL to meet the

other CK' in ZT, and let CH and HP be h and k respec-

tively. Then h and k are the coordinates of P referred to

the asymptotes.

Let a be the semi-angle between the asymptotes, so that,

by Art. 313, tan a = â€” ,

sm a cos a

1

and hence , ,

Draw I{]V perpendicular to the transverse axis, and IIP

parallel to the transverse axis, to meet the ordinate PM of

the point P in P.

ASYMPTOTES AS AXES. 297

Then, since PH and HR are parallel respectively to CL

and CM, we have l PHR ^LLCM=a.

Hence GM^ GN+HR=CHcob a + HPco^a

and MP^BP- UN ^ HP sin a - C^sin a

Therefore, since Cilf and MP satisfy the equation (1),

- "we have

Hence, since (A, ^) is any point on the hyperbola, the

required equation is

xy = â€” 4â€” â–

This is often written in the form xy â€” c^, where 4c^

equals the sum of the squares of the semiaxes of the

hyperbola.

Similarly, the equation to the conjugate hyperbola is,

when referred to the asymptotes,

^y = Jâ€”.

329. To find the equation to the tangent at any point

of the hyperbola xy = c^.

Let (cc', y') be any point P on the hyperbola, and

{x", y") a point Q on it, so that we have

^y = c2 (1),

and x"y" = 0^ (2).

The equation to the line PQ is then

2/ -y'=fc|^ (Â»-=Â«') (3).

298 COORDINATE GEOMETRY.

But, by (1) and (2), we have

c'

y"-v' Â»"

x'

& x'-x"

c^

rJi __ y ~ y

-x'

JU JU tAj ~~~ JU

" x'^'

Hence the equation (3) becomes

c

tAJ %AJ

Let now the point Q be taken indefinitely near to P, so

that x' â€” X ultimately, and therefore, by Art. 149, FQ

becomes the tangent at P.

Then (4) becomes

y

The required equation is therefore

x'i/ + xy â€” Ixy â€” 2g^ (5).

The equation (5) may also be written in the form

% + K-^ -^ (6).

X y

330. The tangent at any point of a hyperbola cuts off a triangle

of constant area from the asymptotes, and the portion of it intercepted

hetioeen the asymptotes is bisected at the point of contact.

Take the asymptotes as axes and let the equation to the hyperbola

be xy = c^.

The tangent at any point P is â€” + â€” , = 2.

X y

This meets the axes in the points [2x' , 0) and (0, 2?/').

If these points be L and U, and the centre be C, we have

GL = 'ix', 2in(iGL' = 2y'.

If 2a be the angle between the asymptotes, the area of the triangle

LGL' = ^CL . OL' sin 2a = 2a;'2/'sin 2a= â€” ^â€” . 2 sinacos a = a6.

(Art. 328.)

Also, since L is the point (2a;', 0) and L' is (0, 2y'), the middle

point of LL' is (x', y'), i.e. the point of contact P.

ASYMPTOTES AS AXES. 299

331. As in Art. 274, the polar of any point (x^, y^

with respect to the curve can be shewn to be

Since, in general, the point (aj^, y^ does not lie on the

curve the equation to the polar cannot be put into the form

(6) of Art. 329.

332. The equation to the normal at the point {x\ y')

is y-y' =^m{xâ€” x'), where m is chosen so that this line is

perpendicular to the tangent

y 2c

y = - ,x + â€”r'

^ X X

If 0) be the angle between the asymptotes we then

obtain, by Art. 93,

x' â€” y cos 0)

m^â€” -, ,

y â€” X cos 0)

so that the required equation to the normal is

y (if' â€” X cos w) â€” X (x â€” y cos w) = y"^ â€” x'.

Also cos w = cos 2a = cos a â€” sm a =r â€” - â€” ~

L 0^ + 6 .

If the hyperbola be rectangular, then co = 90Â°, and the

equation to the normal becomes xx â€” yy' â€” x'^ â€” y'^.

333. Equation referred to the asymptotes.

One Variable.

The equation xy = c^ is clearly satisfied by the substitu-

tion x = ct and y = - .

Hence, for all values of t, the point whose coordinates

are ict, - j lies on the curve, and it may be called the point

The tangent at the point "^" is by Art. 329,

X

300 COORDINATE GEOMETRY

Also the normal is, by the last article,

c

1

c

y (l â€” f cos iji)â€”x {f â€” cos (o) == - (1 - ^*),

or, when the hyperbola is rectangular,

The equations to the tangents at the points ^'t" and " t"

are

-+yt^=. 2c, and - + yt^= 2c,

h h

and hence the tangents meet at the point

Vi + ^2 ' ^1 + tj '

The line joining " t^" and " ^2/' which is the polar of this

point, is therefore, by Art. 331,

X + yt^t^ = c(ti + 1^.

This form also follows by writing down the equation

to the straight line joining the points

(ct,, g and (ct,, Q

334. Ex. 1. If a rectangular hyperbola circumscribe a triangle,

it also passes through the orthocentre of the triangle.

Let the equation to the curve referred to its asymptotes be

a:y = c'^ (1).

Let the angular points of the triangle be P, Q, and R, and let their

coordinates be

j-espectively.

As in the last article, the equation to QR is

x + yt2ts=c{t^ + ts).

The equation to the straight line, through P perpendicular to QR^

is therefore

h

I.e.

y + ctihh=hh[f + jjj] (2)-

ASYMPTOTES AS AXES. 301

Similarly, the equation to the straight line through Q perpendicular

to RP is

2/ + cÂ«iU3=Â¥ir^ + j-7T] (^)-

The common point of (2) and (3) is clearly

{~4j^ "'''''') ^'^'

and this is therefore the orthocentre.

But the coordinates (4) satisfy (1). Hence the proposition.

Also if ( ct^,-\ he the orthocentre of the points " ij," " t^" and

" ig," we have t^t^t^t^â€” - 1.

Ex. 2. If a circle and the rectangular hyperbola (xy = c^ meet in

the four points "ti," "f2>" "^3>" ^nd ^^t^^" prove that

(1) tiÂ«2^3Â«4=l,

(2) the centre of mean position of the four points bisects the

distance between the centres of the two curves,

and (3) the centre of the circle through the points "ij," "f2Â»" "^g" ^^

Let the equation to the circle be

x^ + y^-2gx-^y + k = 0,

so that its centre is the point {g,f).

Any point on the hyperbola is (ct,jj. If this lie on the circle,

we have cH^ + -^-2gct-2f- + k = 0,

so that i4 - 2^ t^+-t^ - ^t +1 = (1).

c c^ c ^ '

If t^ , fg, *3, and ^4 be the roots of this equation, we have, by Art. 2,

hhhh='^ (2),

h + h + t, + t,=^ (3),

2f

and *2M4 + *3*4*1 + *4M2+M2*3= (4).

Dividing (4) by (2), we have

11112/

302 COORDINATE GEOMETRY.

The centre of the mean position of the four points,

i. e. the point || {t^ + *2 + Â«3 + Â«4). | (^- + ^ + ^ + l^f ,

is therefore the point ( k j ^ ) ? ^^^ tliis is the middle point of the line

joining (0, 0) and (^,/)

1

Also, since i^ = -^-^-^ , we have

i-icgi's

= l{t, + h + h+^), and /=|(l + l + i + MA).

Again, since ^1*2*3*4 = 15 "^^ have product of the abscissae of the

four points = product of their ordinates = c^.

EXAMPLES. XXXVIII.

a^ + b^

1. Prove that the foci of the hyperbola xy= â€” j â€” - are given by

2. Shew that two concentric rectangular hyperbolas, whose axes

meet at an angle of 45Â°, cut orthogonally.

3. A straight line always passes through a fixed point; prove

that the locus of the middle point of the portion of it, which is

intercepted between two given straight lines, is a hyperbola whose

asymptotes are parallel to the given lines.

4. If the ordinate NP at any point P of an ellipse be produced to

Q, so that NQ is equal to the subtangent at P, prove that the locus of

^ is a hyperbola.

5. From a point P perpendiculars PM and PN are drawn to two

straight lines OM and ON. If the area OMPN be constant, prove

that the locus of P is a hyperbola.

6. A variable line has its ends on two lines given in position awd

passes through a given point ; prove that the locus of a point which

divides it in any given ratio is a hyperbola.

7. The coordinates of a point are a tan (^ + a) and 6tan(^ + j8),

where 6 is variable ; prove that the locus of the point is a hyperbola.

8. A series of circles touch a given straight line at a given point.

Prove that the locus of the pole of a given straight line with regard to

these circles is a hyperbola whose asymptotes are respectively a

parallel to the first given straight line and a perpendicular to the

second.

[EXS. XXXVIII.] THE HYPEKBOLA. EXAMPLES. 303

9. If a right-angled triangle be inscribed in a rectangular hyper-

bola, prove that the tangent at the right angle is the perpendicular

upon the hypothenuse.

10. In a rectangular hyperbola, prove that all straight lines, which

subtend a right angle at a point P on the curve, are parallel to the

normal at P.

11. Chords of a rectangular hyperbola are at right angles, and

they subtend a right angle at a fixed point ; prove that they inter-

sect on the polar of 0.

12. Prove that any chord of a rectangular hyperbola subtends

angles which are equal or supplementary (1) at the ends of a perpen-

dicular chord, and (2) at the ends of any diameter.

13. In a rectangular hyperbola, shew that the angle between a

chord PQ and the tangent at P is equal to the angle which PQ

subtends at the other end of the diameter through P.

14. Show that the normal to the rectangular hyperbola xy â€” c- at

the point "t" meets the curve again at a point "t"' such that

iÂ¥=-l.

15. If Pj, P25 ^^cl P3 be three points on the rectangular hyperbola

xy = c^, whose abscissae are x-,^, x^ , and x^ , prove that the area of the

triangle P^P^P^ is

^ Â«^-i J;o wo

and that the tangents at these points form a triangle whose area is

2^2 (^2~^3) (^3~^l) (^l~^2)

(^2 + %) (^3 + ^1) {^1 + ^2) "

16. Find the coordinates of the points of contact of common

tangents to the two hyperbolas

x^-y^=Sa^ and xy = 2a^.

17. The transverse axis of a rectangular hyperbola is 2c and the

asymptotes are the axes of coordinates ; shew that the equation of the

chord which is bisected at the point (2c, 3c) is Sx + 2y = 12c.

18. Prove that the portions of any line which are intercepted

between the asymptotes and the curve are equal.

19. Shew that the straight lines drawn from a variable point on

the curve to any two fixed points on it intercept a constant distance on

either asymptote.

20. Shew that the equation to the director circle of the conic

xy = c^is x^ + 2xy cos (>)+y^ = 4ie^ cos (a.

21. Prove that the asymptotes of the hyperbola xy = hx + 1cy are

x = k and y = h.

304 COORDINATE GEOMETRY. [EXS.

22. Shew that the straight line y = mx + c\/ - m always touches the

hyperbola xy = c'^, and that its point of contact is I /-â€” - , of - 711

23. Prove that the locus of the foot of the perpendicular let fall

from' the centre upon chords of the rectangular hyperbola xy = c^

which subtend half a right angle at the origin is the curve

24. A tangent to the parabola x^ = 4a?/ meets the hyperbola xy = k^

in two points P and Q. Prove that the middle point of PQ hes on a

parabola.

25. If a hyperbola be rectangular, and its equation be xy = c^,

prove that the locus of the middle points of chords of constant length

M is {x'^ + 2/^) {xy - c^) = d^xy.

26. Shew that the pole of any tangent to the rectangular hyper-

bola xy^c^, with respect to the circle x^ + y^ = a^, lies on a concentric

and similarly placed rectangular hyperbola.

27. Prove that the locus of the poles of all normal chords of the

rectangular hyperbola xy = c^ is the curve

{a;2- 2/2)2 + 4c2^i/ = 0.

28. Any tangent to the rectangular hyperbola 4:xy = ah meets the

o o

ellipse â€” + â€” â€ž= 1 in the points P and Q ; prove that the normals at P

^ a?" h^

and Q to the ellipse meet on a fixed diameter of the ellipse.

29. Prove that triangles can be inscribed in the hyperbola xy = c^,

whose sides touch the parabola y^=4iax.

30. A. point moves on the given straight line y=mx; prove that

the locus of the foot of the perpendicular let fall from the centre upon

its polar with respect to the ellipse -2 + ^ = '^ is a rectangular

hyperbola, one of whose asymptotes is the diameter of the ellipse

which is conjugate to the given straight line.

31. A quadrilateral circumscribes a hyperbola; prove that the

straight line joining the middle points of its diagonals passes through

the centre of the curve.

32. A, B, C, and D are the points of intersection of a circle and a

rectangular hyperbola. If AB pass through the centre of the hyper-

bola, prove that CD passes through the centre of the circle.

33. If a circle and a rectangular hyperbola meet in four points P,

Q, r] and S, shew that the orthocentres of the triangles QRS, RSP^

SPQ, and PQR^iiso lie on a circle.

Prove also that the tangents to the hyperbola at R and S meet

in a point which lies on the diameter of the hyperbola which is at

right angles to PQ.

XXXVIII.] THE HYPERBOLA. EXAMPLES. 305

34. A series of hyperbolas is drawn, having for asymptotes the

principal axes of an ellipse; shew that the common chords of the

hyperbolas and the ellipse are all parallel to one of the conjugate

diameters of the ellipse.

35. A circle, passing through the centre of a rectangular hyperbola,

cuts the curve in the points A, B, G, and D ; prove that the circum-

circle of the triangle formed by the tangents at A, B, and C goes

through the centre of the hyperbola, and has its centre at the point

of the hyperbola which is diametrically opposite to D.

36. Given five points on a circle of radius a; prove that the

centres of the rectangular hyperbolas, each passing through four of

these points, all lie on a circle of radius - .

follows from the fact that the asymptote is a tangent,

whose point of contact happens to lie at infinity, or it may

be proved directly.

For

GF= CS cos FCS =CS.%= jÂ¥+Â¥ . -.-^^ - a.

Also Z being the foot of the directrix, we have

GA^= CS.cz, (Art. 295)

and hence CF^ = CS . CZ, i.e. CS : CF :: CF : CZ.

By Euc. YI. 6, it follows that l CZF= l CFS= a right

angle, and hence that F lies on the directrix.

Hence the perpendiculars from the foci on either asymptote

meet it in the sam,e points as the corresponding directrix,

and the common points of intersection lie on the auxiliary

circle.

318. Equilateral or Rectangular Hyperbola.

In this curve (Art. 310) the quantities a and b are equal.

The equations to the asymptotes are therefore y=.^x, i.e.

they are inclined at angles =t 45Â° to the axis of x, and hence

they are at right angles. Hence the hyperbola is generally

called a rectangular hyperbola.

319. Conjugate Hyperbola. The hyperbola which

has BB' as its transverse axis, and A A' as its conjugate

axis, is said to be the conjugate hyperbola of the hyperbola

whose transverse and conjugate axes are respectively AA'

and BB'.

Thus the hyperbola

Â¥ a'~^ ^ ^'

is conjugate to the hyperbola

^-^ = 1 (2).

a' b' ^ ^

Just as in Art. 313, the equation to the asymptotes of

F~a'

11^ ex?

X-

THE HYPERBOLA. CONJUGATE DIAMETERS. 289

which, by the same article, is the equation to the asymp-

totes of (2).

Thus a hyperbola and its conjugate have the sam.e

asymptotes.

The conjugate hyperbola is the dotted curve in the

figure of Art. 323.

320. Intersections of a hyperbola with a pair of con-

jugate diameters.

The straight line y = m^x intersects the hyperbola

a^ b'

in points whose abscissae are given by

i.e. by the equation a^ â€” â€” â€” - .

^ 0^ â€” a^m^

The points are therefore real or imaginary, according as

a^TYi^ is < or > 6^,

i.e. according as

m^ is numerically < or > - (1),

a

i.e. according as the inclination of the straight line to the

axis of X is less or greater than the inclination of the

asymptotes.

Now, by Art. 308, the straight lines y - tyi^x and y = m^x

are conjugate diameters if

^' /ox

Â»^1^2 = ^ (2).

Hence one of the quantities m^ and m^ must be less

than - and the other greater than - .

a a

Let mj be < -, so that, by (1), the straight line y = m^x

meets the hyperbola in real points.

L. 19

290 COOEDINATE GEOMETRY.

Then, by (2), m^ must be > - , so that, by (1), the straight

CL

line y = m^x will meet the hyperbola in imaginary points.

It follows therefore that only one of a pair of conjugate

diameters meets a hyperbola in real points.

321. If a pair of diameters he conjugate with respect

to a hyperbola, they will he conjugate with respect to its con-

jugate hyperhola.

Eor the straight lines y = tyi^ and y = m^ are conjugate

with respect to the hyperbola

a^ y"

a

-|.-1 (1).

if ^i'^2==-2 (2).

Cv

Now the equation to the conjugate hyperbola only

differs from (1) in having â€” a^ instead of a^ and - h^ instead

of h% so that the above pair of straight lines will be con-

jugate with respect to it, if

'^â„¢= = l-^ = a^ (^)-

But the relation (3) is the same as (2).

Hence the proposition.

322. If a pair of diameters he conjugate with respect

to a hyperhola, one of them meets the hyperhola in real points

and the other meets the conjugate hyperhola in real points.

Let the diameters be 3/ = m^x and y = m^x, so that

h^

m^m^ = â€” .

a^

As in Art. 320 let mi < - , and hence m- > - , so that the

^ a' ^ Â«'

straight line y â€” m^x meets the hyperbola in real points.

Also the straight line y â€” m^x meets the conjugate

hyperbola ^ -^-\ in points whose abscissae are given by

a

a^y

the equation x^ ( -zj A â€” \y i.e. by .t^ =

ra^ a^ â€” h"^'

THE HYPERBOLA. CONJUGATE DIAMETERS. 291

Since mo > - , these abscissae are real.

" a

Hence the proposition.

323. If aiKiir of conjugate diameters meet the hyperbola

and its conjugate in P and D, then (1) CP^ â€” CD^ = a^ â€” 6^,

and (2) the tangents at P, D and the other ends of the

diameters passing through them form a 'parallelogram whose

vertices lie on the asymptotes and whose area is constant.

Let P be any point on the hyperbola â€” â€” ^ = 1 whose

coordinates are (a sec cf), h tan ^).

The equation to the diameter CP is therefore

h tan 4> ^ â€¢ I

y = J x = x . - sm <p.

a sec <^

a

By Art. 308, the equation to the straight line, Avhich

is conjugate to CP, is

b

y ^x â€” . â€” -

a sm <fi

19â€”2

292 COORDINATE GEOMETRY.

This straight line meets the conjugate hyperbola

in the points (a tan <^, h sec ^), and (â€” a tan (j>, â€”h sec ^) so

that D is the point {a tan <^, 6 sec ^).

We therefore have

CP" - a' sec^ Â«^ + 6Han2 </,,

and CZ)^ - a^ tan^ <^ + Â¥ sec^ <^.

Hence

CP2 _CD^^{a''- h^) (sec2 ^ - tan^ </>) = c*2 - 51

Again, the tangents at P and Z> to the hyperbola and

the conjugate hyperbola are easily seen to be

f sind) =^cos d), (1).

a ^ \ /'

If Q^

and 7 sind) = cosd) (2).

a ^ '

These meet at the point

X y cos ^

a h 1 â€” sin Â«^ *

This point lies on the asymptote CL.

Similarly, the intersection of the tangents at P and D '

lies on CL-[^ that of tangents at D' and F' on GL' ^ and

those at D and F' on CZj.

If tangents be therefore drawn at the points where a

pair of conjugate diameters meet a hyperbola and its

conjugate, they form a parallelogram whose angular points

are on the asymptotes.

Again, the perpendicular from C on the straight line (1)

cos ^ ah cos

y

1 1 . , , V*" + a' sin' A

a6 Â«6 ah

sJhHec^<j> + d'tQ.n^4> CD FK'

THE CONJUGATE HYPERBOLA. 293

SO that PK X perpendicular from G on TK ~ ah,

i.e. area of the parallelogram CPKD â€” ah.

Also the areas of the parallelograms CPKD, CDK^P\

CP'K'D', and CB'K;P are all equal.

The area KK^K'K^ therefore = 4a6.

Cor. PK^ CD = D'C = K^P, so that the portion of a

tangent to a hyperbola intercepted between the asymptotes

is bisected at the point of contact.

324. Relation hefweert the equation to the hyperhola,

the equation to its asymptotes, and the equation to the conju-

gate hyperbola.

The equations to the hyperbola, the asymptotes, and the

conjugate hyperbola are respectively

5-F = i â€¢ W-

%-%-' -(2)-

and-^-|^ = -l (3).

a^ 0"

We notice that the equation (2) differs from equation (1)

by a constant, and that the equation (3) differs from (2) by

exactly the same quantity that (2) differs from (1).

If now we transform the equations in any way we

please â€” by changing the origin and directions of the axes â€”

by the most general substitutions of Art. 132 and by

multiplying the equations by any â€” the same â€” constant,

we shall alter the left-hand members of (1), (2), and (3) in

exactly the same way, and the right-hand constants in the

equations will still be constants, and differ in the same way

as before.

Hence, whatever be the form of the equation to a

hyperbola, the equation to the asymptotes only differs from

it by a constant, and the equation to the conjugate

hyperbola differs from that to the asymptotes by the same

constant.

294 COORDINATE GEOMETRY.

325. As an example of the foregoing article, let it be required

to find the asymptotes of the hyperbola

Sx^-5xy-2y^ + 5x+ny-8 = ...(1).

Since the equation to the asymptotes only differs from it by a

constant, it must be of the form

Sx^-5xy-2y^ + 5x + lly + c = (2).

Since (2) represents the asymptotes it must represent two straight

lines. The condition for this is (Art. 116)

3(-2)c + 2.|.-V-(-f)-3(W-(-2)(t)2-c(-t)2=0,

i.e. c=-12.

The equation to the asymptotes is therefore

3a;2 - 5xy - 2tf + 5a; + 11?/ - 12 = 0,

and the equation to the conjugate hyperbola is

3a;2 - 5xy -2i/ + 5x + lly -16 = 0.

326. As another example we see that the equation to any

hyperbola whose asymptotes are the straight lines

Ax + By + G = and A-^x + B-^y + 0-^ = 0,

is {Ax + By + C){A^x + B^y + Gj) = \'' (1),

where X is any constant.

For (1) only differs by a constant from the equation to the

asymptotes, which is

{Ax + By + C){A^x + B^y + G^) = (2).

If in (1) we substitute - A^ for X^ ^e shall have the equation to its

conjugate hyperbola.

It follows that any equation of the form

[Ax + By + C) {A^x + B-^y + Oj) = X^

represents a hyperbola whose asymptotes are

Ax + By + C = 0, and A^x + B^y + G^ = 0.

Thus the equation x{x + y) = a^ represents a hyperbola whose

asymptotes are x = and x + y = 0.

Again, the equation x'^ + 2xy cot 2a - y^ = a'^,

i.e. {x cot a- y) {x ta.n a + y) = a^,

represents a hyperbola whose asymptotes are

X cot a-y = 0, and x tan a + y = 0.

327. It would follow from the preceding articles that the

equation to any hyperbola whose asymptotes are x = and ?/ = is

a;2/ = const.

THE HYPERBOLA. EXAMPLES. 295

The constant could be easily determined in terms of the semi-

transverse and semi-conjugate axes.

In Art. 328 we shall obtain this equation by direct transformation

from the equation referred to the principal axes.

EXAMPLES. XXXVII.

1. Through the positive vertex of the hyperbola a tangent is

drawn; where does it meet the conjugate hyperbola?

2. lie and e' be the eccentricities of a hyperbola and its conjugate,

prove that _ + _=!.

3. Prove that chords of a hyperbola, which touch the conjugate

hyperbola, are bisected at the point of contact.

4. Shew that the chord, which joins the points in which a pair of

conjugate diameters meets the hyperbola and its conjugate, is parallel

to one asymptote and is bisected by the other.

5. Tangents are drawn to a hyperbola from any point on one of

the branches of the conjugate hyperbola; shew that their chord of

contact will touch the other branch of the conjugate hyperbola.

6. A straight line is drawn parallel to the conjugate axis of a

hyperbola to meet it and the conjugate hyperbola in the points P and

Q ; shew that the tangents at P and Q meet on the curve

6^'

4x^

and that the normals meet on the

axis

of a;.

7. From a point G on the transverse axis GL is drawn perpen-

dicular to the asymptote, and GP a normal to the curve at P. Prove

that LP is parallel to the conjugate axis.

8. Find the asymptotes of the curve 2x'^ + 5xy + 2ij^ -\-4:X + 5y = 0,

and find the general equation of all hyperbolas having the same

asymptotes.

9. Find the equation to the hyperbola, whose asymptotes are the

straight lines x + 2y + 3 = 0, and Sx + 4y + 5 = 0, and which passes

through the point (1,-1).

Write down also the equation to the conjugate hyperbola.

10. In a rectangular hyperbola, prove that CP and CD are equal,

and are inclined to the axis at angles which are complementary.

296

COORDINATE GEOMETRY. [ExS. XXXVII.]

11. C is the centre of the hyperbola -3 - t2 = 1 and the tangent at

any point P meets the asymptotes in the points Q and R. Prove that

the equation to the locus of the centre of the circle circumscribing

the triangle CQB is 4 {a^x^ - b^) = (a^ + b^f.

12. A series of hyperbolas is drawn having a common transverse

axis of length 2a. Prove that the locus of a point P on each hyper-

bola, such that its distance from the transverse axis is equal to its

distance from an asymptote, is the curve {x^-y^)'^=4:X^{x^-a^).

328. To jind the equation to a hyperbola referred to its

asymptotes.

Let P be any point on the hyperbola, whose equation

referred to its axes is

,2 '" ^

.(1).

a" 52

Draw PH parallel to one asymptote CL to meet the

other CK' in ZT, and let CH and HP be h and k respec-

tively. Then h and k are the coordinates of P referred to

the asymptotes.

Let a be the semi-angle between the asymptotes, so that,

by Art. 313, tan a = â€” ,

sm a cos a

1

and hence , ,

Draw I{]V perpendicular to the transverse axis, and IIP

parallel to the transverse axis, to meet the ordinate PM of

the point P in P.

ASYMPTOTES AS AXES. 297

Then, since PH and HR are parallel respectively to CL

and CM, we have l PHR ^LLCM=a.

Hence GM^ GN+HR=CHcob a + HPco^a

and MP^BP- UN ^ HP sin a - C^sin a

Therefore, since Cilf and MP satisfy the equation (1),

- "we have

Hence, since (A, ^) is any point on the hyperbola, the

required equation is

xy = â€” 4â€” â–

This is often written in the form xy â€” c^, where 4c^

equals the sum of the squares of the semiaxes of the

hyperbola.

Similarly, the equation to the conjugate hyperbola is,

when referred to the asymptotes,

^y = Jâ€”.

329. To find the equation to the tangent at any point

of the hyperbola xy = c^.

Let (cc', y') be any point P on the hyperbola, and

{x", y") a point Q on it, so that we have

^y = c2 (1),

and x"y" = 0^ (2).

The equation to the line PQ is then

2/ -y'=fc|^ (Â»-=Â«') (3).

298 COORDINATE GEOMETRY.

But, by (1) and (2), we have

c'

y"-v' Â»"

x'

& x'-x"

c^

rJi __ y ~ y

-x'

JU JU tAj ~~~ JU

" x'^'

Hence the equation (3) becomes

c

tAJ %AJ

Let now the point Q be taken indefinitely near to P, so

that x' â€” X ultimately, and therefore, by Art. 149, FQ

becomes the tangent at P.

Then (4) becomes

y

The required equation is therefore

x'i/ + xy â€” Ixy â€” 2g^ (5).

The equation (5) may also be written in the form

% + K-^ -^ (6).

X y

330. The tangent at any point of a hyperbola cuts off a triangle

of constant area from the asymptotes, and the portion of it intercepted

hetioeen the asymptotes is bisected at the point of contact.

Take the asymptotes as axes and let the equation to the hyperbola

be xy = c^.

The tangent at any point P is â€” + â€” , = 2.

X y

This meets the axes in the points [2x' , 0) and (0, 2?/').

If these points be L and U, and the centre be C, we have

GL = 'ix', 2in(iGL' = 2y'.

If 2a be the angle between the asymptotes, the area of the triangle

LGL' = ^CL . OL' sin 2a = 2a;'2/'sin 2a= â€” ^â€” . 2 sinacos a = a6.

(Art. 328.)

Also, since L is the point (2a;', 0) and L' is (0, 2y'), the middle

point of LL' is (x', y'), i.e. the point of contact P.

ASYMPTOTES AS AXES. 299

331. As in Art. 274, the polar of any point (x^, y^

with respect to the curve can be shewn to be

Since, in general, the point (aj^, y^ does not lie on the

curve the equation to the polar cannot be put into the form

(6) of Art. 329.

332. The equation to the normal at the point {x\ y')

is y-y' =^m{xâ€” x'), where m is chosen so that this line is

perpendicular to the tangent

y 2c

y = - ,x + â€”r'

^ X X

If 0) be the angle between the asymptotes we then

obtain, by Art. 93,

x' â€” y cos 0)

m^â€” -, ,

y â€” X cos 0)

so that the required equation to the normal is

y (if' â€” X cos w) â€” X (x â€” y cos w) = y"^ â€” x'.

Also cos w = cos 2a = cos a â€” sm a =r â€” - â€” ~

L 0^ + 6 .

If the hyperbola be rectangular, then co = 90Â°, and the

equation to the normal becomes xx â€” yy' â€” x'^ â€” y'^.

333. Equation referred to the asymptotes.

One Variable.

The equation xy = c^ is clearly satisfied by the substitu-

tion x = ct and y = - .

Hence, for all values of t, the point whose coordinates

are ict, - j lies on the curve, and it may be called the point

The tangent at the point "^" is by Art. 329,

X

300 COORDINATE GEOMETRY

Also the normal is, by the last article,

c

1

c

y (l â€” f cos iji)â€”x {f â€” cos (o) == - (1 - ^*),

or, when the hyperbola is rectangular,

The equations to the tangents at the points ^'t" and " t"

are

-+yt^=. 2c, and - + yt^= 2c,

h h

and hence the tangents meet at the point

Vi + ^2 ' ^1 + tj '

The line joining " t^" and " ^2/' which is the polar of this

point, is therefore, by Art. 331,

X + yt^t^ = c(ti + 1^.

This form also follows by writing down the equation

to the straight line joining the points

(ct,, g and (ct,, Q

334. Ex. 1. If a rectangular hyperbola circumscribe a triangle,

it also passes through the orthocentre of the triangle.

Let the equation to the curve referred to its asymptotes be

a:y = c'^ (1).

Let the angular points of the triangle be P, Q, and R, and let their

coordinates be

j-espectively.

As in the last article, the equation to QR is

x + yt2ts=c{t^ + ts).

The equation to the straight line, through P perpendicular to QR^

is therefore

h

I.e.

y + ctihh=hh[f + jjj] (2)-

ASYMPTOTES AS AXES. 301

Similarly, the equation to the straight line through Q perpendicular

to RP is

2/ + cÂ«iU3=Â¥ir^ + j-7T] (^)-

The common point of (2) and (3) is clearly

{~4j^ "'''''') ^'^'

and this is therefore the orthocentre.

But the coordinates (4) satisfy (1). Hence the proposition.

Also if ( ct^,-\ he the orthocentre of the points " ij," " t^" and

" ig," we have t^t^t^t^â€” - 1.

Ex. 2. If a circle and the rectangular hyperbola (xy = c^ meet in

the four points "ti," "f2>" "^3>" ^nd ^^t^^" prove that

(1) tiÂ«2^3Â«4=l,

(2) the centre of mean position of the four points bisects the

distance between the centres of the two curves,

and (3) the centre of the circle through the points "ij," "f2Â»" "^g" ^^

Let the equation to the circle be

x^ + y^-2gx-^y + k = 0,

so that its centre is the point {g,f).

Any point on the hyperbola is (ct,jj. If this lie on the circle,

we have cH^ + -^-2gct-2f- + k = 0,

so that i4 - 2^ t^+-t^ - ^t +1 = (1).

c c^ c ^ '

If t^ , fg, *3, and ^4 be the roots of this equation, we have, by Art. 2,

hhhh='^ (2),

h + h + t, + t,=^ (3),

2f

and *2M4 + *3*4*1 + *4M2+M2*3= (4).

Dividing (4) by (2), we have

11112/

302 COORDINATE GEOMETRY.

The centre of the mean position of the four points,

i. e. the point || {t^ + *2 + Â«3 + Â«4). | (^- + ^ + ^ + l^f ,

is therefore the point ( k j ^ ) ? ^^^ tliis is the middle point of the line

joining (0, 0) and (^,/)

1

Also, since i^ = -^-^-^ , we have

i-icgi's

= l{t, + h + h+^), and /=|(l + l + i + MA).

Again, since ^1*2*3*4 = 15 "^^ have product of the abscissae of the

four points = product of their ordinates = c^.

EXAMPLES. XXXVIII.

a^ + b^

1. Prove that the foci of the hyperbola xy= â€” j â€” - are given by

2. Shew that two concentric rectangular hyperbolas, whose axes

meet at an angle of 45Â°, cut orthogonally.

3. A straight line always passes through a fixed point; prove

that the locus of the middle point of the portion of it, which is

intercepted between two given straight lines, is a hyperbola whose

asymptotes are parallel to the given lines.

4. If the ordinate NP at any point P of an ellipse be produced to

Q, so that NQ is equal to the subtangent at P, prove that the locus of

^ is a hyperbola.

5. From a point P perpendiculars PM and PN are drawn to two

straight lines OM and ON. If the area OMPN be constant, prove

that the locus of P is a hyperbola.

6. A variable line has its ends on two lines given in position awd

passes through a given point ; prove that the locus of a point which

divides it in any given ratio is a hyperbola.

7. The coordinates of a point are a tan (^ + a) and 6tan(^ + j8),

where 6 is variable ; prove that the locus of the point is a hyperbola.

8. A series of circles touch a given straight line at a given point.

Prove that the locus of the pole of a given straight line with regard to

these circles is a hyperbola whose asymptotes are respectively a

parallel to the first given straight line and a perpendicular to the

second.

[EXS. XXXVIII.] THE HYPEKBOLA. EXAMPLES. 303

9. If a right-angled triangle be inscribed in a rectangular hyper-

bola, prove that the tangent at the right angle is the perpendicular

upon the hypothenuse.

10. In a rectangular hyperbola, prove that all straight lines, which

subtend a right angle at a point P on the curve, are parallel to the

normal at P.

11. Chords of a rectangular hyperbola are at right angles, and

they subtend a right angle at a fixed point ; prove that they inter-

sect on the polar of 0.

12. Prove that any chord of a rectangular hyperbola subtends

angles which are equal or supplementary (1) at the ends of a perpen-

dicular chord, and (2) at the ends of any diameter.

13. In a rectangular hyperbola, shew that the angle between a

chord PQ and the tangent at P is equal to the angle which PQ

subtends at the other end of the diameter through P.

14. Show that the normal to the rectangular hyperbola xy â€” c- at

the point "t" meets the curve again at a point "t"' such that

iÂ¥=-l.

15. If Pj, P25 ^^cl P3 be three points on the rectangular hyperbola

xy = c^, whose abscissae are x-,^, x^ , and x^ , prove that the area of the

triangle P^P^P^ is

^ Â«^-i J;o wo

and that the tangents at these points form a triangle whose area is

2^2 (^2~^3) (^3~^l) (^l~^2)

(^2 + %) (^3 + ^1) {^1 + ^2) "

16. Find the coordinates of the points of contact of common

tangents to the two hyperbolas

x^-y^=Sa^ and xy = 2a^.

17. The transverse axis of a rectangular hyperbola is 2c and the

asymptotes are the axes of coordinates ; shew that the equation of the

chord which is bisected at the point (2c, 3c) is Sx + 2y = 12c.

18. Prove that the portions of any line which are intercepted

between the asymptotes and the curve are equal.

19. Shew that the straight lines drawn from a variable point on

the curve to any two fixed points on it intercept a constant distance on

either asymptote.

20. Shew that the equation to the director circle of the conic

xy = c^is x^ + 2xy cos (>)+y^ = 4ie^ cos (a.

21. Prove that the asymptotes of the hyperbola xy = hx + 1cy are

x = k and y = h.

304 COORDINATE GEOMETRY. [EXS.

22. Shew that the straight line y = mx + c\/ - m always touches the

hyperbola xy = c'^, and that its point of contact is I /-â€” - , of - 711

23. Prove that the locus of the foot of the perpendicular let fall

from' the centre upon chords of the rectangular hyperbola xy = c^

which subtend half a right angle at the origin is the curve

24. A tangent to the parabola x^ = 4a?/ meets the hyperbola xy = k^

in two points P and Q. Prove that the middle point of PQ hes on a

parabola.

25. If a hyperbola be rectangular, and its equation be xy = c^,

prove that the locus of the middle points of chords of constant length

M is {x'^ + 2/^) {xy - c^) = d^xy.

26. Shew that the pole of any tangent to the rectangular hyper-

bola xy^c^, with respect to the circle x^ + y^ = a^, lies on a concentric

and similarly placed rectangular hyperbola.

27. Prove that the locus of the poles of all normal chords of the

rectangular hyperbola xy = c^ is the curve

{a;2- 2/2)2 + 4c2^i/ = 0.

28. Any tangent to the rectangular hyperbola 4:xy = ah meets the

o o

ellipse â€” + â€” â€ž= 1 in the points P and Q ; prove that the normals at P

^ a?" h^

and Q to the ellipse meet on a fixed diameter of the ellipse.

29. Prove that triangles can be inscribed in the hyperbola xy = c^,

whose sides touch the parabola y^=4iax.

30. A. point moves on the given straight line y=mx; prove that

the locus of the foot of the perpendicular let fall from the centre upon

its polar with respect to the ellipse -2 + ^ = '^ is a rectangular

hyperbola, one of whose asymptotes is the diameter of the ellipse

which is conjugate to the given straight line.

31. A quadrilateral circumscribes a hyperbola; prove that the

straight line joining the middle points of its diagonals passes through

the centre of the curve.

32. A, B, C, and D are the points of intersection of a circle and a

rectangular hyperbola. If AB pass through the centre of the hyper-

bola, prove that CD passes through the centre of the circle.

33. If a circle and a rectangular hyperbola meet in four points P,

Q, r] and S, shew that the orthocentres of the triangles QRS, RSP^

SPQ, and PQR^iiso lie on a circle.

Prove also that the tangents to the hyperbola at R and S meet

in a point which lies on the diameter of the hyperbola which is at

right angles to PQ.

XXXVIII.] THE HYPERBOLA. EXAMPLES. 305

34. A series of hyperbolas is drawn, having for asymptotes the

principal axes of an ellipse; shew that the common chords of the

hyperbolas and the ellipse are all parallel to one of the conjugate

diameters of the ellipse.

35. A circle, passing through the centre of a rectangular hyperbola,

cuts the curve in the points A, B, G, and D ; prove that the circum-

circle of the triangle formed by the tangents at A, B, and C goes

through the centre of the hyperbola, and has its centre at the point

of the hyperbola which is diametrically opposite to D.

36. Given five points on a circle of radius a; prove that the

centres of the rectangular hyperbolas, each passing through four of

these points, all lie on a circle of radius - .

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