Online Library → S. L. (Sidney Luxton) Loney → The elements of coordinate geometry → online text (page 22 of 26)

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i!I=0 and E = 0.

Hence If a conic circumscribe a quadrilateral^ the ratio of the

p7'oduct of the perpendiculars from any point P of the conic upon two

opposite sides of the quadrilateral to the product of the perpendiculars

from P upon the other tioo sides is the same for all positions of P.

384. Equations to the conic sections passing through

the intersections of a conic and two

given straight lines.

Let *S' = be the equation to the

given conic.

Let u-0 and v = be the equa-

tions to the two given straight lines

where

u = ax + hy â– \- Cj

and V = ax + Vy -f- c' .

Let the straight line ^^ = meet the conic /S' = in the

points P and P, and let v = meet it in the points Q and T.

The equation to any conic which passes through the

points P, Q^ R, and T will be of the form

S^X.u.v (1).

For (1) is satisfied by the coordinates of any point

which lies both on S â€” and on u-O] for its coordinates

on being substituted in (1) make both its members zero.

But the points P and R are the only points which lie

both on S â€” and on u = 0.

The equation (1) therefore denotes a conic passing

through P and R.

Similarly it goes through the intersections of ^S' = and

-y = 0, i. e. through the points Q and T.

THE EQUATION S=XuV. 359

Thus (1) represents some conic going through the four

points P, Q, B, and T.

Also (1) represents any conic going through these four

points. For the quantity A. may be so chosen that it shall

go through any fifth point, or to make it satisfy any fifth

condition* also five conditions completely determine a conic

section.

"Ex.. Find the equation to the conic which passes through the point

(1, 1) and also through the intersections of the conic

tvith the straight lines 2x-y - 5 = and Bx + y -11 = 0. Find also

the paradolas passing through the same points.

The equation to the required conic must by the last article be of

the form

x^ + 2xy + 5y^-7x-8y + Q = X {2x -y - 5) {Sx + y -11) ... {1).

This passes through the point (1, 1) if

l + 2 + 5-7-8 + 6=X{2-l-5) (3 + 1-11), i.e. i{ \=-^\.

The required equation then becomes

28{x^+2xy + 5y^-7x-8y + 6) + {2x-y-5) {3x + y -11) = 0,

i.e. 34x2 + 55xy + l'62if - 233a; - 218y + 223 = 0.

The equation to the required parabola will also be of the form (1),

i.e.

x^{l-Q\)+xy{2 + \)+if{5 + \)-x{7-SU)-y{8 + 6\) + Q-55\ = 0.

This is a parabola (Art. 357) if (2 + X)2 = 4 (1 - 6X) (5 + X),

I.e. if X=|[-12Â±4V101.

Substituting these values in (1), we have the required equations.

385. Particular cases of the equation

S = Xuv.

I. Let ^t = and v = intersect on the curve, i.e. in

the figure of Art. 384 let the

points P and Q coincide.

The conic S = Xuv then goes

through two coincident points

at P and therefore touches the

original conic at P as in the

figure.

II. Let ^6 = and v =

coincide, so that v = u.

360

COORDINATE GEOMETRY.

In this case the point T also moves up to coincidence

with R and the second conic

touches the original conic at both ^^ - ''" ~^ \

the points P and B,.

The equation to the second

conic now becomes S=\v?.

When a conic touches a second

conic at each of two points, the

two conies are said to have double

contact with one another.

The two conies S = \v? and /S' = therefore have double

contact with one another, the straight line 16 = passing

through the two points of contact.

As a particular case we see that if ?// â€” 0, v - 0, and

^f; = be the equations to three straight lines then the

equation vw = \v? represents a conic touching the conic

'DW = where u = meets it, i. e. it is a conic to which

v = and w

contact.

are tangents and .u = is the chord of

tl=rO

III. Let u = be a tangent to the original conic.

In this case the two points P

and P coincide, and the conic

S=-Xuv touches S=0 where u=0

touches it, and v = is the equa-

tion to the straight line joining

the other points of intersection of

the two conies.

If, in addition, v â€” goes

through the point of contact of w = 0, we have the equation

to a conic which goes through three coincident points at P,

the point of contact of u = ; also the straight line

joining P to the other point of intersection of the two

conies is v = 0.

IV. Finally, let v = and u = coincide and be

tangents at P. The equation S = Xu^ now represents a

conic section passing through four coincident points at the

point where u-0 touches S = 0.

LINE AT INFINITY. 861

386. Line at infinity. We have shewn, in Art.

60, that the straight line, whose equation is

().x + 0.y + G = 0,

is altogether at an infinite distance. This straight line is

called The Line at Infinity. Its equation may for brevity

be written in the form C = 0.

We can shew that parallel lines meet on the line at

infinity.

For the equations to any two parallel straight lines

are

Ax + By + C =0 (1),

and Ax + By+G' = (2).

Now (2) may be written in the form

Ax + By + C + ^ ~^ (0 . a; + . 3/ + (7) = 0,

and hence, by Art. 97, we see that it passes through the

intersection of (1) and the straight line

0.x + 0.y + C = 0.

Hence (1), (2), and the line at infinity meet in a point.

387. Geometrical rtieaning of the equation

S=Xu (1),

where X is a constant^ ayid u = is the equation of a straight

line.

The equation (1) can be written in the form

/S' = Xz* X (0 . Â£c + . 2/ + 1),

and hence, by Art. 384, represents a conic passing through

the intersection of the conic Sâ€”^ with the straight lines

u = ^ and 0.Â£c + 0.2/ + l = 0.

Hence (1) passes through the intersection of aS'=0 with

the line at infinity.

Since aS' = and S = \u have the same intersections with

362 COORDINATE GEOMETRY.

the line at infinity, it follows that these two conies have

their asjTiiptotes in the same direction.

Particular Case. Let

S = x^ + y^ â€” a^,

so that S = represents a circle.

Any other circle is

x^ + y^- 2gx â€” 2 ft/ + c = 0,

i.e. x^ + y^ â€” a^=2gx+2f'i/ â€” a^ â€” c,

so that its equation is of the form *S' = Xil

It therefore follows that any two circles must be looked

upon as intersecting the line at infinity in the same two

(imaginary) points. These imaginary points are called the

Circular Points at Infinity.

388. Geometrical meaning of the equation S = \ where

\ is a constant.

This equation can be written in the form

S^\{0.x + 0.y+lY,

and therefore, by Art. 385, has double contact with S =

where the straight line .x + Q .y +1 =0 meets it, i.e. the

tangents to the two conies at the points where they meet

the line at infinity are the same.

The conies S=0 and S â€” X therefore have the same

(real or imaginary) asymptotes.

Particular Case. Let S -=^0 denote a circle. Then

S â€” X (being an equation which differs from S â€”0 only in

its constant term) represents a concentric circle.

Two concentric circles must therefore be looked upon as

touching one another at the imaginary points where they

meet the Line at Infinity.

Two concentric circles thus have double contact at the

Circular Points at Infinity.

EXAMPLES. 363

EXAMPLES. XLIII.

1. What is the geometrical meaning of the equations S â€” \. T,

and S = u^ + hu, where ;S = is the equation of a conic, T = is the

equation of a tangent to it, and w = is the equation of any straight

line ?

2. If the major axes of two conies be parallel, prove that the

four points in which they meet are concyclic.

3. Prove that in general two parabolas can be drawn to pass

through the intersections of the conies

ax^ + ^hxy + ly^-^-^gx+^fy+c^Q

and a'x- + 2h'xy + I'y"^ + 2g'x + 2fy + c' = 0,

and that their axes are at right angles if h {a' - V) = h' (a-b).

4. Through the extremities of two focal chords of an ellipse a

conic is described ; if this conic pass through the centre of the ellipse,

prove that it will cut the major axis in another fixed point.

5. Through the extremities of a normal chord of an ellipse a

circle is drawn such that its other common chord passes through the

centre of the ellipse. Prove that the locus of the intersection of

these common chords is an elHpse similar to the given ellipse. If the

eccentricity of the given ellipse be ,^2 (>y2 - 1), prove that the two

ellipses are equal.

6. If two rectangular hyperbolas intersect in four points A, B, G,

and D, prove that the circles described on AB and CD as diameters

cut one another orthogonally.

7. A circle is drawn through the centre of the rectangular

hyperbola xy = c^ to touch the curve and meet it again in two points ;

prove that the locus of the feet of the perpendicular let fall from the

centre upon the common chord is the hyperbola 4^xy = c'^.

8. If a circle touch an ellipse and pass through its centre, prove

that the rectangle contained by the perpendiculars from the centre of

the ellipse upon the common tangent and the common chord is

constant for all points of contact.

9. From a point T whose coordinates are {x', y') a pair of

tangents TP and TQ are drawn to the parabola y'^ = 4:ax; prove that

the liae joining the other pair of points in which the circumcircle of

the triangle TPQ meets the parabola is the polar of the point

(2a -a;', y'), and hence that, if the circle touch the parabola, the line

FQ touches an equal parabola.

10. Prove that the equation to the circle, having double contact

with the ellipse â€” â€ž + ^ = 1 at the ends of a latus rectum, is

x^ + i/-2ae^x = a^{l-e^ - e^).

SQ"^ COORDINATE GEOMETRY. [EXS. XLIII.]

11. Two circles have double contact "with a conic, their chords of

contact being parallel. Prove that the radical axis of the two circles

is midway between the two chords of contact.

12. If a circle and an ellipse have double contact with one another,

prove that the length of the tangent drawn from any point of the

ellipse to the circle varies as the distance of that point from the

chord of contact.

13. Two conies, A and B, have double contact with a third conic

C. Prove that two of the common chords of A and B, and their

chords of contact with G, meet in a point.

14. Prove that the general equation to the ellipse, having double

contact with the circle x^ + y^=a^ and touching the axis of x at the

origin, is c^x^ + (a- + c^) y^ - 2ahy = 0.

15. A rectangular hyperbola has double contact with a fixed

central conic. If the chord of contact always passes through a fixed

point, prove that the locus of the centre of the hyperbola is a circle

passing through the centre of the fixed conic.

16. A rectangular hyperbola has double contact with a parabola ;

prove that the centre of the hyperbola and the pole of the chord of

contact are equidistant from the directrix of the parabola.

389. To find the equation of the jyair of tangents that

can he drawn from any 'point {x, y') to the general conic

<l> (x, y) = ao(f + 2hxy + hy"^ + 2gx + 2fy + c = 0.

Let T be the given point (x, y'), and let P and R be the

points where the tangents from

T touch the conic. /

The equation to PR is there-

fore ^^ = 0,

where u = (ax' + hy -\-g)x

+ {hx + hy +/) y + gx +fy + c.

The equation to any conic

which touches ^S' = at both of

the points P and R is

S=\u\ (Art. 385),

^. e. ao(? + ^hxy + hy^ + '^gx + Ify + c

â€” X \{chx + hy + ^) cc -f- {hx +hy' +f)y + gx +fy' + c]^

(!)â€¢

Now the pair of straight lines TP and TR is a conic

I

DIRECTOR CIRCLE. 365

section which touches the given conic at P and R and

which also goes through the point T.

Also we can only draw one conic to go through five

points, viz. T^ two points at P, and two points at R.

If then we find X so that (1) goes through the point T^

it must represent the two tangents TP and TR.

The equation (1) is satisfied by od and y' if

ax"^ + Ihxy' + hy"^ + ^gx + ^fy + c

- X \ax'' + IJixy + hij"' + 2^a^' + 2fy + c]^,

i.e. if ^'=-irr'> â€” ^\-

The required equation (1) then becomes

^ (x, y) [ax^ + 2hxy + by^ + 2gx + 2fy -f c]

= [(ax + hy' +g)x+ {hx +hy' -\-f)y + gx +fy' + cf,

i.e. ^ (x, y) X ^ (x', y) = u2,

where tt = is the equation to the chord of contact.

390. Director circle of a conic given hy the general

equation of the second degree.

The equation to the two tangents from {x ^ y) to the

conic are, by the last article,

^f? \a^ [x', y') â€” (ax + hy' + gY]

+ 2xy [hcj) (x, y') - (ax' + hy + g) (hx + by' +/)]

+ 2/' [b4> (x, y) - (hx + by' +fY\ + other terms = 0...(1).

If (x', y') be a point on the director circle of the conic,

the two tangents from it to the conic are at right angles.

Now (1) represents two straight lines at right angles if

the sum of the coefficients of x^ and y^ in it be zero,

i.e. if (a + b)<t> (x',y') - (ax + hy' + gf - (hx' + by' +ff = 0.

Hence the locus of the point (x', y) is

(a + b) (ax^ + 2hxy + by^ + 2gx + 2fy + c)

â€” (ax + hy + gf â€” (hx + by +fY â€” 0,

i.e. the circle whose equation is

(x' + f) (ab - A^) + 2x (bg -fh) + 2y (af- gh)

366 COORDINATE GEOMETRY.

Cor. If the given conic be a parabola, then ah = h^,

and the locus becomes a straight line, viz. the directrix of

the parabola. (Art. 211.)

391. The equation to the director circle may also be obtained in

another manner. For it is a circle, whose centre is at the centre of

the conic, and the square of whose radius is equal to the sum of the

squares of the semi-axes of the conic.

The centre is, Art. 352, the point (%^^ , ^^'H ) .

\ab - Ir ab - h^J

Also, if the equation to the conic be reduced to the form

ax^ + 2hxy + hy^ + c' = 0,

and if a and /3 be its semi-axes, we have, (Art. 364,)

i l-^il^ A 1 _ ah-li'^

^ + ^- _c" ^"^^ ^2 - ^^2-'

.1.1 T â€¢ . o ^o â€”{a + h) g'

so that, by division, o?-^p^ = â€” - â€” ^2 â€¢

The equation to the required circle is therefore

gh-afY_ {a + b)c'

i

{"-w^^'^iy

ah ~ h^J ab - h^

{a + h) (abc + 2fgh - af - bg^ - ch^)

{ab - hP-f

(Art. 352).

392. The equation to the (imaginary) tangents drawn

from the focus of a conic to touch the conic satisfies the

analytical condition for being a circle.

Take the focus of the conic as origin, and let the axis of

X be perpendicular to its directrix, so that the equation to

the latter may be written in the form x-{-h = 0.

The equation to the conic, e being its eccentricity, is

therefore x^ + y^= e^ {x + ky,

i.e. jK2(l-e2) + 2/2-2e'^x-e2P^0.

The equation to the pair of tangents drawn from the

origin is therefore, by Art. 389,

[x^ (1 - e^) + 2^2 _ 2e%x - e^P] [- e'k'''] = [- e^kx - e'^JcJ,

i.e. a^ (1 â€” e^) + 2/^ â€” 2e^kx â€” e^k^ - â€” e^[x + k]^,

i.e. x' + y^ = (1).

Here the coefficients of x^ and y"^ are equal and the

coefficient of xy is zero.

FOCI OF THE GENERAL CONIC. 367

However the axes and origin of coordinates be changed,

it follows, on making the substitutions of Art. 129, that in

(1) the coefficients of o(? and y"^ will still be equal and the

coefficient of xy zero.

Hence, whatever be the conic and however its equation

may be written, the equation to the tangents from the focus

always satisfies the analytical conditions for being a circle.

393. To find the foci of the conic given hy the general

equation of the second degree

ax^ + ^hxy + hy'^ + "Igx + ^fy + c = 0.

Let (aj', 2/') be a focus. By the last article the equation

to the pair of tangents drawn from'it satisfies the conditions

for being a circle.

The equation to the pair of tangents is

^ (^', y) [Â«^ + ^hxy + hy'^ + Igx + Ify + c]

= \x {ax' + hy' + g) + y (hx' + by' +/) + (gx' +fy + c)]^.

In this equation the coefficients of xF and y^ must be

equal and the coefficient of xy must be zero.

We therefore have

acf> (x\ y') â€” (ax + hy' + gY = hcf> (x', y) â€” {lix + hy +y* )^,

and 7i0 (a?', y'^ = (ax' + hy' + g) (Jix + by' +f),

i.e.

{ax' + hy' + gf - {hx + by' +ff _ {ax' + hy' + g) {hx' + by' +/)

a â€” b h

= <!>Â¥,&;) â– â€¢â€¢â€¢â€¢w-

These equations, on being solved, give the foci.

Cor. Since the directrices are the polars of the foci,

we easily obtain their equations.

394. The equations (4) of the previous article give, in general,

four values for x' and four corresponding values for y'. Two of these

would be found to be real and two imaginary.

In the case of the ellipse the two imaginary foci lie on the minor

axis. That these imaginary foci exist follows from Art. 247, by

writing the standard equation in the form

368 COORDINATE GEOMETRY.

This shews that the imaginary point {0, sj^^-o^] is a focus, the

imaginary line y . =0 is a directrix, and that the correspond-

ing eccentricity is the imaginary quantity

/&2_a2

Similarly for the hyperbola, except that, in this case, the eccen-

tricity is real.

In the ease of the parabola, two of the foci are at infinity and are

imaginary, whilst a third is at infinity and is real.

395. Ex. 1. Find the focus of the parabola

16a;2 - 2ixy + %2 - 80a; - 140?/ -1- 100=0.

The focus is given by the equations

{IQx' - 1 2y' - 40)^ - ( - 12a;^ + 9y' - 70 )^

rj

_ {16x' - 12y ^ - 40) ( - 12a;^ + 9y' - 70)

~ -12

= lQx'^-24:xY + 9y'^-80x'-U0y' + 100 (1).

The first pair of equation (1) give

12 (16a;' - 12y' - 40)2 + 7 (ig^/ _ i2y' _ 40) ( - 12a;' + 9i/ - 70)

-12 (-12a;' -1-9?/' -70)2=0,

i.e. {4 (16a;' - 12y' - 40) - 3 ( - 12a;' + 9y' - 70)}

X { 3 (16a;' - 12^/' - 40) 4- 4 ( - 12ic' -h %' - 70) } = 0,

i.e. (100a;'-75?/' + 50)x(-400) = 0,

so that y = â€” ^ â€” .

We then have 16a;' - 12^' - 40 = - 48,

and - 12a;' -[-V- 70 =-64.

The second pair of equation (1) then gives

48 X 64

_ ^oxÂ»^ ^ ^, ^^g^, _ ^2^, _ 40) + 2/' ( - 12a;' + 9?/' - 70) - 40a;' - 70^' + 100

= - 48a;' - 64?/' - 40a;' - 70?/' + 100

= - 88a;' -134?/'-}- 100,

^^n 00 , 536a;' -I- 268 ,^^

i.e. -256= -88a;' ~- -t-100,

o

so that a;' =1, and then 2/' =2.

The focua is therefore the point (1, 2).

AXES OF THE GENERAL CONIC. 369

In the case of a parabola, we may also find the equation to the

directrix, by Art. 390, and then find the coordinates of the focus,

which is the pole of the directrix.

Ex. 2. Find the foci of the conic

55a;2 - SOxy + 39^2 - 40a; - 24^/ - 464=0.

The foci are given by the equation

(55a;^ - 15y' - 20)^ - ( - 15a;^ + S9t/ - 12)2

16

_ {55c(/ - 15y' - 20) ( - 15x' + 39?/^ - 12)

-15

= 55a;'2 - 30a;y + 39?/'2 - 40a;' - 24^/' - 464 (1).

The first pair of equations (1) gives

15 (55a;' - 15^' - 20)2 + ig (55^/ _ 15^' _ 20) ( - 15a;' + 39?/' - 12)

- 15 ( - 15a;' + S9y' - 12)2 ^ q,

i.e. {5 (55a;' - 15y' - 20) - 3 ( - 15a;' + 39^' - 12) }

{3 (55a;' - 15?/' - 20) + 5 ( - 15a;' + 39?/' - 12)} = 0,

i.e. (5a;'-3?/'-l)(3a;' + 5?/'-4) = 0.

, 5a;' - 1

â€¢â€¢â€¢ ^ - 3" (2)'

3a;' -4

or y'= â€” 5 - (3).

Substituting this first value of y in the second pair of equation (1),

we obtain

o/;/o / 1X2 340a;'2- 340a;' -1355

- 25 (2a;' - 1)2= ,

giving a;' = 2 or - 1, Hence from (2) y' â€” B or - 2.

On substituting the second value of y' in the same pair of equation

(1), we finally have

2a;'2-2a;' + 13 = 0,

the roots of which are imaginary.

We should thus obtain two imaginary foci which would be found

to lie on the minor axis of the conic section. The real foci are

therefore the points (2, 3) and ( - 1, - 2).

396. Equation to the axes of the general

conic.

By Art. 393, the equation

(ax + hy + gf â€” (hx + by +fy _ {ax â– \-hy-\-g) {hx + hy +/)

a â€” b h

(1)

represents some conic passing through the foci.

L. 24

370

COORDINATE GEOMETRY.

But, since it could be solved as a quadratic equation to

ax + hy + a . ^ ^ ^ . i , t

give 7 j^ â€” ^, it represents two straight lines.

ihoc "T" oy ~r J

The equation (1) therefore represents the axes of the

general conic.

397. To find the length of the straight lines drawn

through a given point in a given direction to vieet a given

conic.

Let the equation to the conic be

<^ (aj, y) = ax^ + 2hxy + hy- + 2gx + Ify + c = . . .(1).

Let F be any point (a;', y')^ and through it let there be

drawn a straight line at an angle Q

with the axis of x to meet the

curve in Q and Q'.

The coordinates of any point

on this line distant r from P

are

Â£c^ + r cos Q and y' + r sin 0.

(Art. 86.)

If this point be on (1), we

have

a{x' + r cos &f + 2A {x + r cos 0) [y' + r sin 0) + b (y' + r sin 0)^

+ 2g {x + r cos 0) + 2/ {y + r sin ^) + c = 0,

i.e.

r^ [a cos2 e + 2h cos ^ sin 6* + & sin^ 0]

+ 2r [(ax + hy' + g) cos d + {hx + by +/) sin ^] + <^ (a;', 3/') =

^ ;-(2).

For any given value of this is a quadratic equation in

r, and therefore for any straight line drawn at an inclina-

tion it gives the values of FQ and PQ'.

If the two values of r given by equation (2) be of

opposite sign, the points Q and Q' lie on opposite sides

of P.

If P be on the curve, then <f) (x, y) is zero and one value

of T obtained from (2) is zero.

Y

R

l>rr

p

C

) X

INTERCEPTS ON LINES. 371

398. If two chords PQQ' and PRR' he drawn in given

directions through any pohit P to meet the curve in Q, Q' and

R^ R' respectively, the ratio of the rectangle PQ . PQ' to the

rectangle PR . PR' is the same for all points, o^nd is therefore

equal to the ratio of the squares of the diameters of the conic

which are drawn in the given directions.

The values of PQ and PQ' are given by the equation of

the last article, and therefore

PQ . PQ' = product of the roots

^ ^ {^\ y) /jx

a cos^ 6 + 2h cos sin ^ + 6 sin^ 6'"^ ''

So, if PRR' be drawn at an angle & to the axis, we have

PR pp'= 9 {^^ y) /o\

a cos^ 0' + 2A cos 0' sin 6' -\-h sin^ 0'"'^ ^'

On dividing (1) by (2), we have

PQ . PQ' _ a cos^ 6' + 2h cos 0' sin 0' + 1} sin^ 6'

PR . PR' ~ a cos^ e-\-2h cos ^ sin ^ + 6 sin^ 6 '

The right-hand member of this equation does not contain

x' or y', i. e. it does not depend on the position of P but only

on the directions and $\

The quantity ^ ' p> is therefore the same for all

positions of P.

In the particular case when P is at the centre of the

CO'"^

conic this ratio becomes ,^ â– , , where C is the centre and CQ'

OR

and CR" are parallel to the two given directions.

Cor. If Q and Q' coincide, and also R and R', the two

lines PQQ' and PRR become the tangents from P, and the

above relation then gives

PQ^C(r' . PQ _ CQ"

PR' ~ OR'"' ' ""â€¢ ^' PR ~ CR" â–

Hence, If two tangents be drawn from a point to a conic,

their lengths are to one another in the ratio of the parallel

semi-diameters of the conic.

24â€”2

372

COORDINATE GEOMETRY.

399. If PQQ' and P^QiQi ^^ ^'^^ chords drawn in

parcdlel directions from, two points P aiid F^ to meet a conic

in Q and Q', and Q^ and Qi, resjyectively, then the ratio of

the rectangles PQ . PQ' and PiQi . PiQi is independent of the

direction of the chords.

For, if P and P^ be respectively the points (x', y') and

{x", y")y and 6 be the angle that each chord makes with

the axis, we have, as in the last article,

^ (^', y)

PQ.PQ'=^

and P^Q,,P^Q;^

a cos2 e-\-2h cos 6' sin (9 + 6 sin^ '

<i> K, y")

so that

a cos^ 6 -\-'2ih cos 6 sin ^ + 6 sin^ '

PQ.PQ' <i>{x,y')

PiQi-P^Qi ^{x",y"y

400. If a circle and a conic section cut one another in four 'points,

the straight line joining one pair of points of intersection and the

straight line joining the other pair are equally inclined to the axis of

the conic.

For (Fig. Art. 397) let the circle and conic intersect in the four

points Q, Q' and B, B' and let QQ' and RB' meet in P.

But, since Q, Q', B, and B' are four points on a circle, we have

PQ . PQ' = PB . PB'. [Euc. III. 36, Cor.]

.-. CQ" = CB".

Also in any conic equal radii from the centre are equally inclined

to the axis of the conic.

Hence GQ" and CB", and therefore PQQ' and PBB\ are equally

inclined to the axis of the conic.

401. To shew that any chord of a conic is cut har-

monically by the curve, any point on

the chord^ and the pjolar of this point

with respect to the conic.

Take the point as origin, and let

the equation to the conic be

ao(? + ^hxy + hy"^ + 2gx + 2/3/ + c -

(!)>

HARMONIC PROPERTY OF THE POLAR. 373

or, in polar coordinates,

r^ {a cos^ + 2h cos ^ sin ^ + 6 sin^ 6) + 2r {g cos 6 +/sin 6) + c^Q,

i.e.

c. -^+2 .-. (gcosO +/sin 0)

+ a cos^ + 2h cos sin + b sin^O=^ 0.

Hence, if the chord OFF' be drawn at an angle to OX,

we have

TT^i + yc-T^, â€” sum of the roots of this equation in -

OF OF r

g cos 6 +y sin 6

c

Let -ff be a point on this chord such that

2___1_ J

OF ~ of"" OF' '

Then, if OR = p, we have

2 ^cos^+ysin^

P ^

so that the locus of F is

g . p cos +/. p sin + c â€” Oj

or, in Cartesian coordinates,

gx+/y + G^O (2).

But (2) is the polar of the origin with respect to the

conic (1), so that the locus of F is the polar of 0.

The straight line FF' is therefore cut harmonically by

Hence If a conic circumscribe a quadrilateral^ the ratio of the

p7'oduct of the perpendiculars from any point P of the conic upon two

opposite sides of the quadrilateral to the product of the perpendiculars

from P upon the other tioo sides is the same for all positions of P.

384. Equations to the conic sections passing through

the intersections of a conic and two

given straight lines.

Let *S' = be the equation to the

given conic.

Let u-0 and v = be the equa-

tions to the two given straight lines

where

u = ax + hy â– \- Cj

and V = ax + Vy -f- c' .

Let the straight line ^^ = meet the conic /S' = in the

points P and P, and let v = meet it in the points Q and T.

The equation to any conic which passes through the

points P, Q^ R, and T will be of the form

S^X.u.v (1).

For (1) is satisfied by the coordinates of any point

which lies both on S â€” and on u-O] for its coordinates

on being substituted in (1) make both its members zero.

But the points P and R are the only points which lie

both on S â€” and on u = 0.

The equation (1) therefore denotes a conic passing

through P and R.

Similarly it goes through the intersections of ^S' = and

-y = 0, i. e. through the points Q and T.

THE EQUATION S=XuV. 359

Thus (1) represents some conic going through the four

points P, Q, B, and T.

Also (1) represents any conic going through these four

points. For the quantity A. may be so chosen that it shall

go through any fifth point, or to make it satisfy any fifth

condition* also five conditions completely determine a conic

section.

"Ex.. Find the equation to the conic which passes through the point

(1, 1) and also through the intersections of the conic

tvith the straight lines 2x-y - 5 = and Bx + y -11 = 0. Find also

the paradolas passing through the same points.

The equation to the required conic must by the last article be of

the form

x^ + 2xy + 5y^-7x-8y + Q = X {2x -y - 5) {Sx + y -11) ... {1).

This passes through the point (1, 1) if

l + 2 + 5-7-8 + 6=X{2-l-5) (3 + 1-11), i.e. i{ \=-^\.

The required equation then becomes

28{x^+2xy + 5y^-7x-8y + 6) + {2x-y-5) {3x + y -11) = 0,

i.e. 34x2 + 55xy + l'62if - 233a; - 218y + 223 = 0.

The equation to the required parabola will also be of the form (1),

i.e.

x^{l-Q\)+xy{2 + \)+if{5 + \)-x{7-SU)-y{8 + 6\) + Q-55\ = 0.

This is a parabola (Art. 357) if (2 + X)2 = 4 (1 - 6X) (5 + X),

I.e. if X=|[-12Â±4V101.

Substituting these values in (1), we have the required equations.

385. Particular cases of the equation

S = Xuv.

I. Let ^t = and v = intersect on the curve, i.e. in

the figure of Art. 384 let the

points P and Q coincide.

The conic S = Xuv then goes

through two coincident points

at P and therefore touches the

original conic at P as in the

figure.

II. Let ^6 = and v =

coincide, so that v = u.

360

COORDINATE GEOMETRY.

In this case the point T also moves up to coincidence

with R and the second conic

touches the original conic at both ^^ - ''" ~^ \

the points P and B,.

The equation to the second

conic now becomes S=\v?.

When a conic touches a second

conic at each of two points, the

two conies are said to have double

contact with one another.

The two conies S = \v? and /S' = therefore have double

contact with one another, the straight line 16 = passing

through the two points of contact.

As a particular case we see that if ?// â€” 0, v - 0, and

^f; = be the equations to three straight lines then the

equation vw = \v? represents a conic touching the conic

'DW = where u = meets it, i. e. it is a conic to which

v = and w

contact.

are tangents and .u = is the chord of

tl=rO

III. Let u = be a tangent to the original conic.

In this case the two points P

and P coincide, and the conic

S=-Xuv touches S=0 where u=0

touches it, and v = is the equa-

tion to the straight line joining

the other points of intersection of

the two conies.

If, in addition, v â€” goes

through the point of contact of w = 0, we have the equation

to a conic which goes through three coincident points at P,

the point of contact of u = ; also the straight line

joining P to the other point of intersection of the two

conies is v = 0.

IV. Finally, let v = and u = coincide and be

tangents at P. The equation S = Xu^ now represents a

conic section passing through four coincident points at the

point where u-0 touches S = 0.

LINE AT INFINITY. 861

386. Line at infinity. We have shewn, in Art.

60, that the straight line, whose equation is

().x + 0.y + G = 0,

is altogether at an infinite distance. This straight line is

called The Line at Infinity. Its equation may for brevity

be written in the form C = 0.

We can shew that parallel lines meet on the line at

infinity.

For the equations to any two parallel straight lines

are

Ax + By + C =0 (1),

and Ax + By+G' = (2).

Now (2) may be written in the form

Ax + By + C + ^ ~^ (0 . a; + . 3/ + (7) = 0,

and hence, by Art. 97, we see that it passes through the

intersection of (1) and the straight line

0.x + 0.y + C = 0.

Hence (1), (2), and the line at infinity meet in a point.

387. Geometrical rtieaning of the equation

S=Xu (1),

where X is a constant^ ayid u = is the equation of a straight

line.

The equation (1) can be written in the form

/S' = Xz* X (0 . Â£c + . 2/ + 1),

and hence, by Art. 384, represents a conic passing through

the intersection of the conic Sâ€”^ with the straight lines

u = ^ and 0.Â£c + 0.2/ + l = 0.

Hence (1) passes through the intersection of aS'=0 with

the line at infinity.

Since aS' = and S = \u have the same intersections with

362 COORDINATE GEOMETRY.

the line at infinity, it follows that these two conies have

their asjTiiptotes in the same direction.

Particular Case. Let

S = x^ + y^ â€” a^,

so that S = represents a circle.

Any other circle is

x^ + y^- 2gx â€” 2 ft/ + c = 0,

i.e. x^ + y^ â€” a^=2gx+2f'i/ â€” a^ â€” c,

so that its equation is of the form *S' = Xil

It therefore follows that any two circles must be looked

upon as intersecting the line at infinity in the same two

(imaginary) points. These imaginary points are called the

Circular Points at Infinity.

388. Geometrical meaning of the equation S = \ where

\ is a constant.

This equation can be written in the form

S^\{0.x + 0.y+lY,

and therefore, by Art. 385, has double contact with S =

where the straight line .x + Q .y +1 =0 meets it, i.e. the

tangents to the two conies at the points where they meet

the line at infinity are the same.

The conies S=0 and S â€” X therefore have the same

(real or imaginary) asymptotes.

Particular Case. Let S -=^0 denote a circle. Then

S â€” X (being an equation which differs from S â€”0 only in

its constant term) represents a concentric circle.

Two concentric circles must therefore be looked upon as

touching one another at the imaginary points where they

meet the Line at Infinity.

Two concentric circles thus have double contact at the

Circular Points at Infinity.

EXAMPLES. 363

EXAMPLES. XLIII.

1. What is the geometrical meaning of the equations S â€” \. T,

and S = u^ + hu, where ;S = is the equation of a conic, T = is the

equation of a tangent to it, and w = is the equation of any straight

line ?

2. If the major axes of two conies be parallel, prove that the

four points in which they meet are concyclic.

3. Prove that in general two parabolas can be drawn to pass

through the intersections of the conies

ax^ + ^hxy + ly^-^-^gx+^fy+c^Q

and a'x- + 2h'xy + I'y"^ + 2g'x + 2fy + c' = 0,

and that their axes are at right angles if h {a' - V) = h' (a-b).

4. Through the extremities of two focal chords of an ellipse a

conic is described ; if this conic pass through the centre of the ellipse,

prove that it will cut the major axis in another fixed point.

5. Through the extremities of a normal chord of an ellipse a

circle is drawn such that its other common chord passes through the

centre of the ellipse. Prove that the locus of the intersection of

these common chords is an elHpse similar to the given ellipse. If the

eccentricity of the given ellipse be ,^2 (>y2 - 1), prove that the two

ellipses are equal.

6. If two rectangular hyperbolas intersect in four points A, B, G,

and D, prove that the circles described on AB and CD as diameters

cut one another orthogonally.

7. A circle is drawn through the centre of the rectangular

hyperbola xy = c^ to touch the curve and meet it again in two points ;

prove that the locus of the feet of the perpendicular let fall from the

centre upon the common chord is the hyperbola 4^xy = c'^.

8. If a circle touch an ellipse and pass through its centre, prove

that the rectangle contained by the perpendiculars from the centre of

the ellipse upon the common tangent and the common chord is

constant for all points of contact.

9. From a point T whose coordinates are {x', y') a pair of

tangents TP and TQ are drawn to the parabola y'^ = 4:ax; prove that

the liae joining the other pair of points in which the circumcircle of

the triangle TPQ meets the parabola is the polar of the point

(2a -a;', y'), and hence that, if the circle touch the parabola, the line

FQ touches an equal parabola.

10. Prove that the equation to the circle, having double contact

with the ellipse â€” â€ž + ^ = 1 at the ends of a latus rectum, is

x^ + i/-2ae^x = a^{l-e^ - e^).

SQ"^ COORDINATE GEOMETRY. [EXS. XLIII.]

11. Two circles have double contact "with a conic, their chords of

contact being parallel. Prove that the radical axis of the two circles

is midway between the two chords of contact.

12. If a circle and an ellipse have double contact with one another,

prove that the length of the tangent drawn from any point of the

ellipse to the circle varies as the distance of that point from the

chord of contact.

13. Two conies, A and B, have double contact with a third conic

C. Prove that two of the common chords of A and B, and their

chords of contact with G, meet in a point.

14. Prove that the general equation to the ellipse, having double

contact with the circle x^ + y^=a^ and touching the axis of x at the

origin, is c^x^ + (a- + c^) y^ - 2ahy = 0.

15. A rectangular hyperbola has double contact with a fixed

central conic. If the chord of contact always passes through a fixed

point, prove that the locus of the centre of the hyperbola is a circle

passing through the centre of the fixed conic.

16. A rectangular hyperbola has double contact with a parabola ;

prove that the centre of the hyperbola and the pole of the chord of

contact are equidistant from the directrix of the parabola.

389. To find the equation of the jyair of tangents that

can he drawn from any 'point {x, y') to the general conic

<l> (x, y) = ao(f + 2hxy + hy"^ + 2gx + 2fy + c = 0.

Let T be the given point (x, y'), and let P and R be the

points where the tangents from

T touch the conic. /

The equation to PR is there-

fore ^^ = 0,

where u = (ax' + hy -\-g)x

+ {hx + hy +/) y + gx +fy + c.

The equation to any conic

which touches ^S' = at both of

the points P and R is

S=\u\ (Art. 385),

^. e. ao(? + ^hxy + hy^ + '^gx + Ify + c

â€” X \{chx + hy + ^) cc -f- {hx +hy' +f)y + gx +fy' + c]^

(!)â€¢

Now the pair of straight lines TP and TR is a conic

I

DIRECTOR CIRCLE. 365

section which touches the given conic at P and R and

which also goes through the point T.

Also we can only draw one conic to go through five

points, viz. T^ two points at P, and two points at R.

If then we find X so that (1) goes through the point T^

it must represent the two tangents TP and TR.

The equation (1) is satisfied by od and y' if

ax"^ + Ihxy' + hy"^ + ^gx + ^fy + c

- X \ax'' + IJixy + hij"' + 2^a^' + 2fy + c]^,

i.e. if ^'=-irr'> â€” ^\-

The required equation (1) then becomes

^ (x, y) [ax^ + 2hxy + by^ + 2gx + 2fy -f c]

= [(ax + hy' +g)x+ {hx +hy' -\-f)y + gx +fy' + cf,

i.e. ^ (x, y) X ^ (x', y) = u2,

where tt = is the equation to the chord of contact.

390. Director circle of a conic given hy the general

equation of the second degree.

The equation to the two tangents from {x ^ y) to the

conic are, by the last article,

^f? \a^ [x', y') â€” (ax + hy' + gY]

+ 2xy [hcj) (x, y') - (ax' + hy + g) (hx + by' +/)]

+ 2/' [b4> (x, y) - (hx + by' +fY\ + other terms = 0...(1).

If (x', y') be a point on the director circle of the conic,

the two tangents from it to the conic are at right angles.

Now (1) represents two straight lines at right angles if

the sum of the coefficients of x^ and y^ in it be zero,

i.e. if (a + b)<t> (x',y') - (ax + hy' + gf - (hx' + by' +ff = 0.

Hence the locus of the point (x', y) is

(a + b) (ax^ + 2hxy + by^ + 2gx + 2fy + c)

â€” (ax + hy + gf â€” (hx + by +fY â€” 0,

i.e. the circle whose equation is

(x' + f) (ab - A^) + 2x (bg -fh) + 2y (af- gh)

366 COORDINATE GEOMETRY.

Cor. If the given conic be a parabola, then ah = h^,

and the locus becomes a straight line, viz. the directrix of

the parabola. (Art. 211.)

391. The equation to the director circle may also be obtained in

another manner. For it is a circle, whose centre is at the centre of

the conic, and the square of whose radius is equal to the sum of the

squares of the semi-axes of the conic.

The centre is, Art. 352, the point (%^^ , ^^'H ) .

\ab - Ir ab - h^J

Also, if the equation to the conic be reduced to the form

ax^ + 2hxy + hy^ + c' = 0,

and if a and /3 be its semi-axes, we have, (Art. 364,)

i l-^il^ A 1 _ ah-li'^

^ + ^- _c" ^"^^ ^2 - ^^2-'

.1.1 T â€¢ . o ^o â€”{a + h) g'

so that, by division, o?-^p^ = â€” - â€” ^2 â€¢

The equation to the required circle is therefore

gh-afY_ {a + b)c'

i

{"-w^^'^iy

ah ~ h^J ab - h^

{a + h) (abc + 2fgh - af - bg^ - ch^)

{ab - hP-f

(Art. 352).

392. The equation to the (imaginary) tangents drawn

from the focus of a conic to touch the conic satisfies the

analytical condition for being a circle.

Take the focus of the conic as origin, and let the axis of

X be perpendicular to its directrix, so that the equation to

the latter may be written in the form x-{-h = 0.

The equation to the conic, e being its eccentricity, is

therefore x^ + y^= e^ {x + ky,

i.e. jK2(l-e2) + 2/2-2e'^x-e2P^0.

The equation to the pair of tangents drawn from the

origin is therefore, by Art. 389,

[x^ (1 - e^) + 2^2 _ 2e%x - e^P] [- e'k'''] = [- e^kx - e'^JcJ,

i.e. a^ (1 â€” e^) + 2/^ â€” 2e^kx â€” e^k^ - â€” e^[x + k]^,

i.e. x' + y^ = (1).

Here the coefficients of x^ and y"^ are equal and the

coefficient of xy is zero.

FOCI OF THE GENERAL CONIC. 367

However the axes and origin of coordinates be changed,

it follows, on making the substitutions of Art. 129, that in

(1) the coefficients of o(? and y"^ will still be equal and the

coefficient of xy zero.

Hence, whatever be the conic and however its equation

may be written, the equation to the tangents from the focus

always satisfies the analytical conditions for being a circle.

393. To find the foci of the conic given hy the general

equation of the second degree

ax^ + ^hxy + hy'^ + "Igx + ^fy + c = 0.

Let (aj', 2/') be a focus. By the last article the equation

to the pair of tangents drawn from'it satisfies the conditions

for being a circle.

The equation to the pair of tangents is

^ (^', y) [Â«^ + ^hxy + hy'^ + Igx + Ify + c]

= \x {ax' + hy' + g) + y (hx' + by' +/) + (gx' +fy + c)]^.

In this equation the coefficients of xF and y^ must be

equal and the coefficient of xy must be zero.

We therefore have

acf> (x\ y') â€” (ax + hy' + gY = hcf> (x', y) â€” {lix + hy +y* )^,

and 7i0 (a?', y'^ = (ax' + hy' + g) (Jix + by' +f),

i.e.

{ax' + hy' + gf - {hx + by' +ff _ {ax' + hy' + g) {hx' + by' +/)

a â€” b h

= <!>Â¥,&;) â– â€¢â€¢â€¢â€¢w-

These equations, on being solved, give the foci.

Cor. Since the directrices are the polars of the foci,

we easily obtain their equations.

394. The equations (4) of the previous article give, in general,

four values for x' and four corresponding values for y'. Two of these

would be found to be real and two imaginary.

In the case of the ellipse the two imaginary foci lie on the minor

axis. That these imaginary foci exist follows from Art. 247, by

writing the standard equation in the form

368 COORDINATE GEOMETRY.

This shews that the imaginary point {0, sj^^-o^] is a focus, the

imaginary line y . =0 is a directrix, and that the correspond-

ing eccentricity is the imaginary quantity

/&2_a2

Similarly for the hyperbola, except that, in this case, the eccen-

tricity is real.

In the ease of the parabola, two of the foci are at infinity and are

imaginary, whilst a third is at infinity and is real.

395. Ex. 1. Find the focus of the parabola

16a;2 - 2ixy + %2 - 80a; - 140?/ -1- 100=0.

The focus is given by the equations

{IQx' - 1 2y' - 40)^ - ( - 12a;^ + 9y' - 70 )^

rj

_ {16x' - 12y ^ - 40) ( - 12a;^ + 9y' - 70)

~ -12

= lQx'^-24:xY + 9y'^-80x'-U0y' + 100 (1).

The first pair of equation (1) give

12 (16a;' - 12y' - 40)2 + 7 (ig^/ _ i2y' _ 40) ( - 12a;' + 9i/ - 70)

-12 (-12a;' -1-9?/' -70)2=0,

i.e. {4 (16a;' - 12y' - 40) - 3 ( - 12a;' + 9y' - 70)}

X { 3 (16a;' - 12^/' - 40) 4- 4 ( - 12ic' -h %' - 70) } = 0,

i.e. (100a;'-75?/' + 50)x(-400) = 0,

so that y = â€” ^ â€” .

We then have 16a;' - 12^' - 40 = - 48,

and - 12a;' -[-V- 70 =-64.

The second pair of equation (1) then gives

48 X 64

_ ^oxÂ»^ ^ ^, ^^g^, _ ^2^, _ 40) + 2/' ( - 12a;' + 9?/' - 70) - 40a;' - 70^' + 100

= - 48a;' - 64?/' - 40a;' - 70?/' + 100

= - 88a;' -134?/'-}- 100,

^^n 00 , 536a;' -I- 268 ,^^

i.e. -256= -88a;' ~- -t-100,

o

so that a;' =1, and then 2/' =2.

The focua is therefore the point (1, 2).

AXES OF THE GENERAL CONIC. 369

In the case of a parabola, we may also find the equation to the

directrix, by Art. 390, and then find the coordinates of the focus,

which is the pole of the directrix.

Ex. 2. Find the foci of the conic

55a;2 - SOxy + 39^2 - 40a; - 24^/ - 464=0.

The foci are given by the equation

(55a;^ - 15y' - 20)^ - ( - 15a;^ + S9t/ - 12)2

16

_ {55c(/ - 15y' - 20) ( - 15x' + 39?/^ - 12)

-15

= 55a;'2 - 30a;y + 39?/'2 - 40a;' - 24^/' - 464 (1).

The first pair of equations (1) gives

15 (55a;' - 15^' - 20)2 + ig (55^/ _ 15^' _ 20) ( - 15a;' + 39?/' - 12)

- 15 ( - 15a;' + S9y' - 12)2 ^ q,

i.e. {5 (55a;' - 15y' - 20) - 3 ( - 15a;' + 39^' - 12) }

{3 (55a;' - 15?/' - 20) + 5 ( - 15a;' + 39?/' - 12)} = 0,

i.e. (5a;'-3?/'-l)(3a;' + 5?/'-4) = 0.

, 5a;' - 1

â€¢â€¢â€¢ ^ - 3" (2)'

3a;' -4

or y'= â€” 5 - (3).

Substituting this first value of y in the second pair of equation (1),

we obtain

o/;/o / 1X2 340a;'2- 340a;' -1355

- 25 (2a;' - 1)2= ,

giving a;' = 2 or - 1, Hence from (2) y' â€” B or - 2.

On substituting the second value of y' in the same pair of equation

(1), we finally have

2a;'2-2a;' + 13 = 0,

the roots of which are imaginary.

We should thus obtain two imaginary foci which would be found

to lie on the minor axis of the conic section. The real foci are

therefore the points (2, 3) and ( - 1, - 2).

396. Equation to the axes of the general

conic.

By Art. 393, the equation

(ax + hy + gf â€” (hx + by +fy _ {ax â– \-hy-\-g) {hx + hy +/)

a â€” b h

(1)

represents some conic passing through the foci.

L. 24

370

COORDINATE GEOMETRY.

But, since it could be solved as a quadratic equation to

ax + hy + a . ^ ^ ^ . i , t

give 7 j^ â€” ^, it represents two straight lines.

ihoc "T" oy ~r J

The equation (1) therefore represents the axes of the

general conic.

397. To find the length of the straight lines drawn

through a given point in a given direction to vieet a given

conic.

Let the equation to the conic be

<^ (aj, y) = ax^ + 2hxy + hy- + 2gx + Ify + c = . . .(1).

Let F be any point (a;', y')^ and through it let there be

drawn a straight line at an angle Q

with the axis of x to meet the

curve in Q and Q'.

The coordinates of any point

on this line distant r from P

are

Â£c^ + r cos Q and y' + r sin 0.

(Art. 86.)

If this point be on (1), we

have

a{x' + r cos &f + 2A {x + r cos 0) [y' + r sin 0) + b (y' + r sin 0)^

+ 2g {x + r cos 0) + 2/ {y + r sin ^) + c = 0,

i.e.

r^ [a cos2 e + 2h cos ^ sin 6* + & sin^ 0]

+ 2r [(ax + hy' + g) cos d + {hx + by +/) sin ^] + <^ (a;', 3/') =

^ ;-(2).

For any given value of this is a quadratic equation in

r, and therefore for any straight line drawn at an inclina-

tion it gives the values of FQ and PQ'.

If the two values of r given by equation (2) be of

opposite sign, the points Q and Q' lie on opposite sides

of P.

If P be on the curve, then <f) (x, y) is zero and one value

of T obtained from (2) is zero.

Y

R

l>rr

p

C

) X

INTERCEPTS ON LINES. 371

398. If two chords PQQ' and PRR' he drawn in given

directions through any pohit P to meet the curve in Q, Q' and

R^ R' respectively, the ratio of the rectangle PQ . PQ' to the

rectangle PR . PR' is the same for all points, o^nd is therefore

equal to the ratio of the squares of the diameters of the conic

which are drawn in the given directions.

The values of PQ and PQ' are given by the equation of

the last article, and therefore

PQ . PQ' = product of the roots

^ ^ {^\ y) /jx

a cos^ 6 + 2h cos sin ^ + 6 sin^ 6'"^ ''

So, if PRR' be drawn at an angle & to the axis, we have

PR pp'= 9 {^^ y) /o\

a cos^ 0' + 2A cos 0' sin 6' -\-h sin^ 0'"'^ ^'

On dividing (1) by (2), we have

PQ . PQ' _ a cos^ 6' + 2h cos 0' sin 0' + 1} sin^ 6'

PR . PR' ~ a cos^ e-\-2h cos ^ sin ^ + 6 sin^ 6 '

The right-hand member of this equation does not contain

x' or y', i. e. it does not depend on the position of P but only

on the directions and $\

The quantity ^ ' p> is therefore the same for all

positions of P.

In the particular case when P is at the centre of the

CO'"^

conic this ratio becomes ,^ â– , , where C is the centre and CQ'

OR

and CR" are parallel to the two given directions.

Cor. If Q and Q' coincide, and also R and R', the two

lines PQQ' and PRR become the tangents from P, and the

above relation then gives

PQ^C(r' . PQ _ CQ"

PR' ~ OR'"' ' ""â€¢ ^' PR ~ CR" â–

Hence, If two tangents be drawn from a point to a conic,

their lengths are to one another in the ratio of the parallel

semi-diameters of the conic.

24â€”2

372

COORDINATE GEOMETRY.

399. If PQQ' and P^QiQi ^^ ^'^^ chords drawn in

parcdlel directions from, two points P aiid F^ to meet a conic

in Q and Q', and Q^ and Qi, resjyectively, then the ratio of

the rectangles PQ . PQ' and PiQi . PiQi is independent of the

direction of the chords.

For, if P and P^ be respectively the points (x', y') and

{x", y")y and 6 be the angle that each chord makes with

the axis, we have, as in the last article,

^ (^', y)

PQ.PQ'=^

and P^Q,,P^Q;^

a cos2 e-\-2h cos 6' sin (9 + 6 sin^ '

<i> K, y")

so that

a cos^ 6 -\-'2ih cos 6 sin ^ + 6 sin^ '

PQ.PQ' <i>{x,y')

PiQi-P^Qi ^{x",y"y

400. If a circle and a conic section cut one another in four 'points,

the straight line joining one pair of points of intersection and the

straight line joining the other pair are equally inclined to the axis of

the conic.

For (Fig. Art. 397) let the circle and conic intersect in the four

points Q, Q' and B, B' and let QQ' and RB' meet in P.

But, since Q, Q', B, and B' are four points on a circle, we have

PQ . PQ' = PB . PB'. [Euc. III. 36, Cor.]

.-. CQ" = CB".

Also in any conic equal radii from the centre are equally inclined

to the axis of the conic.

Hence GQ" and CB", and therefore PQQ' and PBB\ are equally

inclined to the axis of the conic.

401. To shew that any chord of a conic is cut har-

monically by the curve, any point on

the chord^ and the pjolar of this point

with respect to the conic.

Take the point as origin, and let

the equation to the conic be

ao(? + ^hxy + hy"^ + 2gx + 2/3/ + c -

(!)>

HARMONIC PROPERTY OF THE POLAR. 373

or, in polar coordinates,

r^ {a cos^ + 2h cos ^ sin ^ + 6 sin^ 6) + 2r {g cos 6 +/sin 6) + c^Q,

i.e.

c. -^+2 .-. (gcosO +/sin 0)

+ a cos^ + 2h cos sin + b sin^O=^ 0.

Hence, if the chord OFF' be drawn at an angle to OX,

we have

TT^i + yc-T^, â€” sum of the roots of this equation in -

OF OF r

g cos 6 +y sin 6

c

Let -ff be a point on this chord such that

2___1_ J

OF ~ of"" OF' '

Then, if OR = p, we have

2 ^cos^+ysin^

P ^

so that the locus of F is

g . p cos +/. p sin + c â€” Oj

or, in Cartesian coordinates,

gx+/y + G^O (2).

But (2) is the polar of the origin with respect to the

conic (1), so that the locus of F is the polar of 0.

The straight line FF' is therefore cut harmonically by

Online Library → S. L. (Sidney Luxton) Loney → The elements of coordinate geometry → online text (page 22 of 26)