Online Library → S. L. (Sidney Luxton) Loney → The elements of coordinate geometry → online text (page 16 of 26)

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X. ae ^ . a

a^ e

i.e. the corresponding directrix.

275. "When the point (iCj, y^ lies outside the ellipse,

the equation to its polar is the same as the equation of the

chord of contact of tangents from it.

When (a^i, y^ is on the ellipse, its polar is the same as

the tangent at it.

As in Art. 215 the polar of (x^^ y^ might have been

defined as the chord of contact of the tangents, real or

imaginary, drawn from it.

276. By a proof similar to that of Art. 217 it can be

shewn that If the polar of P pass through T, then the polar

of T passes through P.

PAIR OF TANGENTS FROM ANY POINT. 251

277. To find the coordinates of the pole of any given

Ixii/e

Ax + By + C=0 (1).

Let (x^, 2/i) be its pole. Then (1) must be the same as

the polar of (a^i, y^)j i.e.

^" + f'-l=0 ..(2).

Comparing (1) and (2), as in Art. 218, the required pole

is easily seen to be

278. To find the equation to the pair of tangents that

can be drawn to the ellipse from the point (x^^j y^.

Let (h, k) be any point on either of the tangents that

can be drawn to the ellipse.

The equation of the straight line joining (A, k) to

(fljj, 2/i) is

^ - Vl / x

k â€” y^ hy. â€” kx.

t.e. y = j â€” ^x+^ ^\

If this straight line touch the ellipse, it must be of the

form

y = mx + s]a^w? + h^. ( Art. 263.)

Hence

rnJ^, and f^^lJl^V = â€žW + Jl

h â€” x^ \ h â€” x^ J

Hence f ^^' " ^â– Y = Â«^ faV + 6'.

\ h-x^ J \h-xj

But this is the condition that the point (A, U) may lie

on the locus

{xy^ - x^yf =a'{y- y^f + b''{x- x^f (1).

This equation is therefore the equation to the required

tangents.

252 COORDINATE GEOMETRY.

It would be found that (1) is equivalent to

/^' ,2/' i\ /^^i' ^3// A _ f^^i ^ 2/2/1 1

279. To Jind the locus of the middle points of parallel

chords of the ellij)se.

Let the chords make with the axis an angle whose

tangent is m, so that the equation to any one of them,

QR, is

y = 7nx + c (1),

where c is different for the different chords.

This straight line meets the ellipse in points whose

abscissae are given by the equation

x^ {mx + cY

a-

-1.

i. e. ^ {a'm' + 6') + la'mcx + a'{c'-b^) = (2).

Let the roots of this equation, i.e. the abscissae of Q

and H, be x^ and x^, and let F, the middle point of QB, be

the point [h, k).

Then, by Arts. 22 and 1, we have

tJC-i *T* t^o

a^mc

C^Vl^ + IP'

(3).

CONJUGATE DIAMETERS. 253

Also V lies on the straight line (1), so that

k = mh + c (4).

If between (3) and (4) we eliminate c, we have

. a^TTb (k â€” tnh)

^" a?rri' + }f '

i. e. hVi = ~ a^mk (5).

Hence the point (A, k) always lies on the straight line

y â€” â€” 5 â€” X (6).

The required locus is therefore the straight line

y â€” 7n-,x, where m, = ^râ€” ,

^'

I.e. mm. â€” -â€ž (7).

280. Equation to the chord ivhose middle point is {h, k).

The required equation is (1) of the foregoing article, where m and

c are given by equations (4) and (5), so that

b^h , a?k^ + y-h'^

m=-^.andc = __^-.

The required equation is therefore

^~ a'k''^ a?k '

i.e. ^^{y-Jc)+~{X-h)=0.

It is therefore parallel to the polar of {h, k).

281. Diameter. Def. The locus of the middle

points of parallel chords of an ellipse is called a diameter,

and the chords are called its double ordinates.

By equation (6) of Art. 279 we see that any diameter

passes through the centre C.

Also, by equation (7), we see that the diameter 7/ â€” m^x

bisects all chords parallel to the diameter y â€” mx, if

62

mm

'-a^ W-

254 COORDINATE GEOMETRY.

But the symmetry of the result (1) shows that, in this

case, the diameter y â€” rax bisects all chords parallel to the

diameter y â€” m^x.

Such a pair of diameters are called Conjugate Diameters.

Hence

Conjugate Diameters. Def. Two diameters are

said to be conjugate when each bisects all chords parallel

to the other.

Two diameters y â€” tnx and y â€” m^x are therefore con-

jugate, if

miXli = â€” â€” g- ,

^ a-*

282. The tangent at the extremity of any diameter is

parallel to the chords which it bisects.

In the Figure of Art. 279 let [x, y') be the point P on

the ellipse, the tangent at which is parallel to the chord

QBj whose equation is

y â€” inx + c (1).

The tangent at the point (a;', y) is

^' + ^'=1 (2).

a-' o" ^ '

Since (1) and (2) are parallel, we have

a-'y

i.e. the point (cc', y^ lies on the straight line

a^m,

But, by Art. 279, this is the diameter which bisects QR

and all chords which are parallel to it.

Cor. It follows that two conjugate diameters CP and

CD are such that each is parallel to the tangent at the

extremity of the other. Hence, given either of these, we

have a geometrical construction for the other.

CONJUGATE DIAMETERS. 255

283. The tangents at the ends of any chord meet on the

diameter which bisects the chord.

Let the equation to the chord QR (Art. 279) be

2/ = ma;+ c (1).

Let T be the point of intersection of the tangents at Q

and Rj and let its coordinates be x^ and y^.

Since QR is the chord of contact of tangents from T^ its

equation is, by Art. 273,

^V^^l (2)

The equations (1) and (2) therefore represent the same

straight line, so that

})%

i.e. {h^ k) lies on the straight line

am,

which, by Art. 279, is the equation to the diameter bisect-

ing the chord QR. Hence T lies on the straight line GP.

284. If the eccentric angles of the ends, P and D, of a

pair of conjugate diameters he <^ and (f>', then <^ and cf>' differ

by a right angle.

Since P is the point [a cos ^, h sin </>), the equation to

CPis

y~x.â€” tan </> (1).

yâ€” X . - tan <^' (2).

a

So the equation to CD is

h

a

These diameters are (Art. 281) conjugate if

6^ , Â¥

â€” tan d> tan ch ~ - â€” ^ ,

a^ ^ ^ a^

i. e. if tan <^ ^^ â€” cot cf}' â€” tan (<^' Â± 90Â°j,

i.e. if ^-^'= + 90Â°.

256

COOEDINATE GEOMETRY.

Cor. 1. The points on the auxiliary circle correspond-

ing to P and D subtend a right angle at the centre.

For if p and d be these points then, by Art. 258, we

have

z.79(7^'-<^ and LdCA'^^'.

Hence

LpGd=^ LdCA' - LpGA' = <f>-<j>' = 90\

Cor. 2. In the figure of Art. 286 if P be the point <fi,

then D is the point ^ + 90Â° and D' is the point <^ - 90Â°.

285. From the previous article it follows that if P be

the point {a cos c(>, b sin <^), then D is the point

{a cos (90Â° + <^), b sin (90Â° + <^)} i- e. (â€” a sin <^, b cos </>).

Hence, if PJ^ and DM be the ordinates of P and i),

we have

iTP CM

and

CJSr MD

a

a

286. If PCP' and BCD' be a pair of conjugate dia-

meierSj then (1) CP^ + CD^ is constant, and (2) the area of

the parallelogram formed by the tangents at the ends of these

diameters is constant.

CONJUGATE DIAMETERS. 257

Let P be the point ^, so that its coordinates are a cos ^

and h sin <^. Then D is the point 90Â° + </>, so that its co-

ordinates are

a cos (90Â° + ^) and h sin (90Â°+ <^),

i.e. â€”a sin ^ and h cos <^.

(1) We therefore have

CP^ = Â«' cos^ <^ + 6' sin^ <^,

and CD'^ - a? sin^ cj> + b"^ cos^ ^,

Hence CP^ + CB' = 0"+ b'^

â€” the sum of the squares of the semi-axes of the ellipse.

(2) Let KLMN be the parallelogram formed by the

tangents at P, D, P', and D'.

By Euc. I. 36j we have

area KLMN = 4 . area CPXD

-4. GU.PK=iCU.CD,

where CU is the perpendicular from C upon the tangent

at P.

Now the equation to the tangent at P is

- cos <l> + T sin (i) â€” 1 = 0,

a

so that (Art. 75) we have

1 ab ab

C (J

/cos^

^</) sin2<^ J a" sin'' <f, + b^ cos' <f> CD

Hence CU.CI) = ab.

Thus the area of the parallelogram KLMN = iab,

which is equal to the rectangle formed by the tangents

at the ends of the major and minor axes.

287. The product of the focal distances of a point P is

equal to the square on the semidiameter parallel to the tangent

at P.

If P be the point (p, then, by Art. 251, we have

SP = a + ae cos tf), and S'P â€”a â€” ae cos ^.

L. 17

258 COORDINATE GEOMETRY.

Hence SP . S'P ^a'~ are' cos' <^

= a^ sin^ <^+b' cos^ <^

288. Ex. If P and D be the ends of conjugate diameters, find

the locus of

(1) the middle point of PD,

(2) the intersection of the tangents at P and D,

and (3) the foot of the perpendicular from the centre upon PD.

P is the point (a cos 0, h sin 0) and D is ( - a sin 0, 6 cos 0).

(1) If (x, y) be the middle point of PD, we have

a cos d>-a sin d) ^ h sin + & cos

^= ^ ^' ^^^ ^ = 2 â€¢

If we eliminate we shall get the required locus. We obtain

2 2

^ + |2=i[(cos - sin 0)2+ (sin + cos 0)2] = ^.

The locus is therefore a concentric and similar ellipse.

[N.B. Two ellipses are similar if the ratios of their axes are the

same, so that they have the same eccentricity.]

(2) The tangents are

-cos0 + vsin0=l,

a b

and - sin0 + rcos0 = l.

a b ^

Both of these equations hold at the intersection of the tangents.

If we eliminate we shall have the equation of the locus of their

intersections.

By squaring and adding, we have

so that the locus is another similar and concentric ellipse.

(3) By Art. 259, on putting 0' = 9OÂ° + 0, the equation to PD is

- cos (45Â° + 0) + 1 sin (45Â° + 0) = cos 45Â°.

Let the length of the perpendicular from the centre be 2^ and let it

make an angle w with the axis. Then this line must be equivalent to

xcosu + ysm(a=p.

CONJUGATE DIAMETERS. 259

Comparing the equations, we have

..^r. X a cos CO cos 45Â° , . , , _â€ž , . & sin w cos 45Â°

cos (45Â° + ^) = , and sin(4oÂ° + <^) =

Hence, by squaring and adding, 2p2=a2cos2 w + ft^sin^w, i.e. the

locus required is the curve

289. Equiconjugate diameters. Let P and D be ex-

tremities of equiconjugate diameters, so that CP^ = CDK

If the eccentric angle of P be 0, we then have

a^ cos^ (f> + b^ sin^ (f> = a^ sin^ (f} + h^ cos^ ^,

giving tan^ ^ = Ij

i.e. ^ = 45Â°, or 135Â°.

The equation to CP is then

h

y = x. â€” tan ^,

*.e. 2/ = Â±-^ (1)>

and that to CD is y~ â€” x â€” cot <^,

a

^.e.

2/ = + -^ (2).

If a rectangle be formed whose sides are the tangents

at Aj A', Â£, and B' the lines (1) and (2) are easily seen to

be its diagonals.

The directions of the equiconjugates are therefore along

the diagonals of the circumscribing rectangle.

The length of each equiconjugate is, by Art. 286,

290. Supplemental chords. Def. The chords

joining any point P on an ellipse to the extremities, P and

P'j of any diameter of the ellipse are called supplemental

chords.

Supplemental chords are parallel to conjugate diatneters.

17â€”2

260 COORDINATE GEOMETRY.

Let P be the point whose eccentric angle is <f>, and JR

and E' the points whose eccentric angles are <^i and

180Â° + <^i.

The equations to FB and FE' are then (Art. 259)

^ oo^ *^ + ^^ + ^ sin ilii-cos ^^1 a^

and

X ^+180Â° + ^! 2/ . ^+180Â°+^_ <^_180Â°-^

â€” cos ;r -r ^ Sm ^;; â€” â€¢ COS -z:

a 2 b 2 2 '

*.e. - - sm ^Hr^ + y cos ^ ^ ^^ = sm ^ ^ ^\ .. (2).

a 2 6 2 2 ^ '

The Â« m " of the straight line (1) = cot tÂ±il .

The " m " of the line (2) = - tan ^^^ .

CO A

If

The product of these " m's " = g , so that, by Art. 281,

the lines FR and P^' are parallel to conjugate diameters.

This proposition may also be easily proved geometrically.

For let V and V be the middle points of PJ2 and TR'.

Since V and C are respectively the middle points of EP and J2E',

the hne OF is parallel to FBI, Similarly CV is parallel to FB.

Since GV bisects PE it bisects all chords parallel to PP, i.e. all

chords parallel to GV. So CF' bisects all chords parallel to GV.

Hence CF and GV are in the direction of conjugate diameters and

therefore PP' and JPB,., being parallel to (7F and GV respectively, are

parallel to conjugate diameters.

CONJUGATE DIAMETERS. 261

291. To find the equation to an ellipse referred to a

pair of conjugate diameters.

Let the conjugate diameters be CP and CD (Fig. Art.

286), whose lengths are a' and h' respectively.

If we transform the equation to the ellipse, referred to

its principal axes, to CP and CD as axes of coordinates,

then, since the origin is unaltered, it becomes, by Art. 134,

of the form

Ax' + 2Hxy-\-By'=l (1).

Now the point P, {a, 0), lies on (1), so that

Aa" = \ (2).

So since Q, the point (0, h'), lies on (1), we have

Bh'^ = 1.

Hence A^-jz-, and B = -=-fâ€ž.

a^ h^

Also, since CP bisects all chords parallel to CD, there-

fore for each value of x we have two equal and opposite

values of y. This cannot be unless 11=0.

The equation then becomes

^+^-=1

Cor. If the axes be the equiconjugate diameters, the

equation is x^ + y^ = a'^. The equation is thus the same in

form as the equation to a circle. In the case of the ellipse

however the axes are oblique.

292. It will be noted that the equation to the ellipse,

when referred to a pair of conjugate diameters, is of the

same form as it is when referred to its principal axes.

The latter are merely a particular case of a pair of conjugate

diameters.

Just as in Art. 262, it may be shewn that the equation

to the tangent at the point (x, y') is

Similarly for the equation to the polar.

262 COORDINATE GEOMETRY.

Ex. If QVQ' be a double ordinate of the diameter CP, and if the

tangent at Q meet CP in T, then CV . CT=CP'^.

If Q be the point [x', y'), the tangent at it is

Putting y = 0, we have

^rj._a'^_GP^

I.e.

i.e. CV.CT=GP^

EXAMPLES. XXXIV.

1. In the ellipse qF + tt = 1> ^^^ ^^^ equation to the chord which

passes through the point (2, 1) and is bisected at that point.

2. Find, with respect to the elHpse 4:X^ + 7y^=8,

(1) the polar of the point ( - J, 1), and

(2) the pole of the straight line 12a; + 7t/ + 16 = 0.

3. Tangents are drawn from the point {3, 2) to the ellipse

x^ + 4:y'^=9. Find the equation to their chord of contact and the

equation of the straight line joining (3, 2) to the middle point of this

chord of contact.

4. Write down the equation of the pair of tangents drawn to the

ellipse Sx^ + 2y^=5 from the point (1, 2), and prove that the angle

between them is tan"^ â€” ^ .

a

5. In the ellipse -s + ^=l, write down the equations to the

diameters which are conjugate to the diameters whose equations are

x-y = 0, x + y = 0, y = ^x, and 2/ = -^-

6. Shew that the diameters whose equations are y + Sx = and

iy-x â€” O, are conjugate diameters of the ellipse 3x^ + 4:y^=5.

7. If the product of the perpendiculars from the foci upon the

polar of P be constant and equal to c^, prove that the locus of P is the

elHpSe 6%2 (^2 + ^2^2^ + g2^4^2^ ^4^4.

8. Shew that the four lines which join the foci to two points P

and Q on an ellipse all touch a circle whose centre is the pole of PQ.

[EXS. XXXIV.] CONJUGATE DIAMETERS. EXAMPLES. 263

9. If the pole of the normal at P lie on the normal at Q, then

shew that the pole of the normal at Q lies on the normal at P.

10. CK is the perpendicular from the centre on the polar of any

point P, and PM is the perpendicular from P on the same polar and

is produced to meet the major axis in L. Shew that (1) CK . PL = b^,

and (2) the product of the perpendiculars from the foci on the polar

= CK.LM.

What do these theorems become when P is on the ellipse ?

11. In the previous question, if PN be the ordinate of P and the

polar meet the axis in T, shew that CL = e^. CN and CT . CN-a^.

12. If tangents TP and TQ be drawn from a point T, whose

coordinates are h and k, prove that the area of the triangle TPQ is

and that the area of the quadrilateral CPTQ is

13. Tangents are drawn to the ellipse from the point

prove that they intercept on the ordinate through the nearer focus a

distance equal to the major axis.

14. Prove that the angle between the tangents that can be drawn

from any point [x^ , y{) to the ellipse is

2ah /â– ~-Â±^^yl

tan~i

x{^ + y-^ -a^-b^

15. If T be the point {x^, y-y), shew that the equation to the

straight lines joining it to the foci, S and S\ is

{x-^y - xy-yf - a\^ {y - i/i)^ = 0.

Prove that the bisector of the angle between these lines also

bisects the angle between the tangents TP and TQ that can be drawn

from T, and hence that

lSTP=lS'TQ.

16. If two tangents to an ellipse and one of its foci be given, prove

that the locus of its centre is a straight line.

17. Prove that the straight lines joining the centre to the inter-

sections of the straight line y=mx+ a/ â€” ^ with the ellipse are

conjugate diameters.

264 COOEDINATE GEOMETRY. [Exs. XXXIV.]

18. Any tangent to an ellipse meets the director circle in -p and d ;

prove that Cp and Gd are in the directions of conjugate diameters of

the ellipse.

19. If CP be conjugate to the normal at Q, prove that GQ is

conjugate to the normal at P.

20. If a fixed straight line parallel to either axis meet a pair of

conjugate diameters in the points K and L, shew that the circle

described on KL as diameter passes through two fixed points on the

other axis.

21. Prove that a chord which joins the ends of a pair of conjugate

diameters of an ellipse always touches a similar ellipse.

22. The eccentric angles of two points P and Q on the ellipse are

01 and 02 ' prove that the area of the parallelogram formed by the

tangents at the ends of the diameters through P and Q is

4a&cosec(0i-02),

and hence that it is least when P and Q are at the end of conjugate

diameters.

23. -A. pair of conjugate diameters is produced to meet the

directrix; shew that the orthocentre of the triangle so formed is at

the focus.

24. If the tangent at any point P meet in the points L and L'

(1) two parallel tangents, or (2) two conjugate diameters,

prove that in each case the rectangle LP . PL' is equal to the square

on the semidiameter which is parallel to the tangent at P.

25. -A. point is such that the perpendicular from the centre on its

polar with respect to the ellipse is constant and equal to c ; shew that

its locus is the elhpse

^2 2/2^1

26. Tangents are drawn from any point on the ellipse -3 + 'rg =1

to the circle x'^ + y'^=r'^ ; prove that the chords of contact are tangents

to the ellipse a^x^^- h^y'^ = r^.

If â€” = -5 + -s , prove that the lines joining the centre to the points

r^ a^ 0^

of contact with the circle are conjugate diameters of the second

ellipse.

27. GP and CD are conjugate diameters of the ellipse ; prove that

the locus of the orthocentre of the triangle GPD is the curve

2 {hh)'^ + aV)^^ (a2 - 62)2 (^2^2 _ ^2^2)2,

28. If circles be described on two semi-conjugate diameters of the

ellipse as diameters, prove that the locus of their second points of

intersection is the curve 2{x^-\-y'^)'^=a^x^ + h^y'^.

FOUR NORMALS TO AN ELLIPSE. 265

293. To prove that, in general, four normals can he

drawn from any point to an ellipse, and that the sum of the

eccentric angles of their feet is equal to an odd m/ultiple of

two right angles.

The normal at any point, whose eccentric angle is ^, is

r-^ = a^-h^^ de^.

cos <p sm <p

If this normal pass through the point (A, ^), we have

J!L_JL=Â«v (1).

COS </) sm <jt> ^ '

For a given point (li, k) this equation gives the

eccentric angles of the feet of the normals which pass

through {h, h).

<k

Let tan ^ = ^, so that

cos d> = , = -:, ^ , and sni cb = = .

l + tan^t lH-t-4 '^*

Substituting these values in (1), we have

ah ^ 7, â€” ok

2/,2

â€” a^e

\-f 2t

i.e. hkf + 2^3 (ah + aV) + 2i5 (ah - a^e^) -hk = ... (2).

Let ^1, ^2) hi ^^(i h ^6 the roots of this equation, so that,

by Art. 2,

^ah + a?e^ , ,

^X + ^2+^3 + ^4 = -2â€” ^^- (3),

kk + ^1^3 + ^1^4 + ^2^3 + ^2^4 + kh =^0 (4),

^ ah â€” a^p} , ,

^2^3^4 + 4^1 + Â«54^li^2 + ^1^3 = - 2 TT (5),

and ^j^2^o^4=:-l (6),

266 COOEDINATE GEOMETRY.

Hence {Trigonometry, Art. 125), we have

tan {h + ^ + il + h\ - ^1-^3 __ ^1 - ^3 _ ^

â€¢â€¢ 2 ='''^+2'

and hence <^i + ^2 + ^3 + </>4 = (2n + 1)??

= an odd multiple of two right angles.

294. We shall conclude the chapter with some ex-

amples of loci connected with the ellipse.

Ex. 1. Find the locus of the intersection of tangents at the ends

of chords of an ellipse, which are of constant length 2c.

Let QR be any such chord, and let the tangents at Q and R meet

in a point P, whose coordinates are {h, k).

Since QR is the polar of P, its equation is

The abscissae of the points in which this straight line meets the

ellipse are given by

\^ a^j ~ h^ V ^y '

L (^ fc^\ _2Â£i

x^ [W W-\ 2xh , Ti^ ^

If x^ and .^2 be the roots of this equation, i.e. the abscissae of Q

and R, we have

^l + ^2-^-2p:;-^2;i;2' and ^1^2 -j2;,2 + ^2/^2-

If 2/i and 2/2 be the ordinates of Q and R, we have from (1)

a2 "^ 62 -â€¢^'

and 2+^=1'

so that, by subtraction,

_ i'h

2/2 ~ 2/1 â€” ^2^ V.^2 ~ ^l/'

THE ELLIPSE. EXAMPLES. 267

The condition of the question therefore gives

Hence the point {h, k) always lies on the curve

^ Ka'^y^J-y &2 "^ aP ) V^ + P~-^j'

which is therefore the locus of P.

Ex. 2. Find the locus (1) of the middle points, and (2) of the poles,

of normal chords of the ellipse.

The chord, whose middle point is {h, k), is parallel to the polar of

(h, k) , and is therefore

i^-h)^,+ {y-k)^^=o (1).

If this be a normal, it must be the same as

aa^sec 6 - by cosec d = a^-h^ (2).

We therefore have

gsec 6 _ - 6 cosec 6 a^-b^

"T % ^ /i2 k-^ '

so that cos^:

a^ b^ a^ b^

and sin^=-

h{a^-b^)

fh^ Jf\

\^ '^ by '

y^'^by

k (a2 - fc2)

Hence, by the elimination of d,

fc2\2

The equation to the required locus is therefore

@4:)(^-^i=' - ^^

e4:r(^3-'-'^>=

Again, if (x^, y-y) be the pole of the normal chord (2), the latter

equation must be equivalent to the equation

^^ + ^^=1 (3).

Comparing (2) and (3), we have

a^sec0_ 6^ cosec ^

^1 vT' '

( a^ W\

so that l = cos2^ + sin2^= â€” 5+â€” 5) â€”

\x^ y^j (a

&2,

1

62)2 â€¢

268 COORDINATE GEOMETRY.

and hence the required locus is

Ex. 3. Chords of the ellipse â€” 2 + ^2 â€” â– 'â– ^^^'"^^/s touch the concentric

and coaxal ellipse -k + ^ = 1; fend the locus of their poles.

Any tangent to the second ellipse is

yz=mx+ ^Ja^m^ + p^ (1).

Let the tangents at the points where it meets the first ellipse meet

in (h, k). Then (1) must be the same as the polar of {h, k) with

respect to the first ellipse, i.e. it is the same as

a^^b'' ^~" ^''^â€¢

Since (1) and (2) coincide, we have

'h~ k~ Ti

Hence m= - ^T, and fja^m^ + jS^ == - .

a^ k

Eliminating m, we have

a4/c2 + P -/^2'

i.e. the point {h, k) lies on the ellipse

^2 1)2

i.e. on a concentric and coaxal ellipse whose semi-axes are â€” and â€”

â– a p

respectively.

EXAMPLES. XXXV.

The tangents drawn from a point P to the ellipse make angles 61

and $2 with the major axis ; find the locus of P when

1. ^1 + ^2 i^ constant (=:2a). [Compare Ex. 1, Art. 235.]

2. tan ^i + tan 62 is constant ( = c).

3. tan ^1 - tan d^ is constant ( = d!).

4. tan^ d-i + tan^ ^o is constant ( = X).

[EXS. XXXV.] THE ELLIPSE. EXAMPLES. 269

Find the locus of the intersection of tangents

5. which meet at a given angle a.

6. if the sum of the eccentric angles of their points of contact

be equal to a constant angle 2a.

7. if the difference of these eccentric angles be 120Â°.

8. if the lines joining the points of contact to the centre be

perpendicular.

9. if the sum of the ordinates of the points of contact be equal to h.

Find the locus of the midSle points of chords of an ellipse

10. whose distance from the centre is the constant length c.

11. which subtend a right angle at the centre.

12. which pass through the given point (/i, Tc).

13. whose length is constant ( = 2c).

14. whose poles are on the auxiliary circle.

15. the tangents at the ends of which intersect at right angles.

16. Prove that the locus of the intersection of normals at the

ends of conjugate diameters is the curve

2 {a?x^ + hhff= {o? - b^ {a^^^ - b^yT-

17. Prove that the locus of the intersection of normals at the ends

of chords, parallel to the tangent at the point whose eccentric angle is

a, is the conic

2 [ax sin a + by cos a) {ax cos a + by sin a) = (a^ - &2)2 g^j^ 2a cos^ 2a.

If the chords be parallel to an equiconjugate diameter, the locus

is a diameter perpendicular to the other equiconjugate.

18. A parallelogram circumscribes the ellipse and two of its

opposite angular points lie on the straight lines x'^ = h^; prove that

the locus of the other two is the conic

^2 y2

(-.:)-â–

19. Circles of constant radius c are drawn to pass through the

ends of a variable diameter of the ellipse. Prove that the locus of

a^ e

i.e. the corresponding directrix.

275. "When the point (iCj, y^ lies outside the ellipse,

the equation to its polar is the same as the equation of the

chord of contact of tangents from it.

When (a^i, y^ is on the ellipse, its polar is the same as

the tangent at it.

As in Art. 215 the polar of (x^^ y^ might have been

defined as the chord of contact of the tangents, real or

imaginary, drawn from it.

276. By a proof similar to that of Art. 217 it can be

shewn that If the polar of P pass through T, then the polar

of T passes through P.

PAIR OF TANGENTS FROM ANY POINT. 251

277. To find the coordinates of the pole of any given

Ixii/e

Ax + By + C=0 (1).

Let (x^, 2/i) be its pole. Then (1) must be the same as

the polar of (a^i, y^)j i.e.

^" + f'-l=0 ..(2).

Comparing (1) and (2), as in Art. 218, the required pole

is easily seen to be

278. To find the equation to the pair of tangents that

can be drawn to the ellipse from the point (x^^j y^.

Let (h, k) be any point on either of the tangents that

can be drawn to the ellipse.

The equation of the straight line joining (A, k) to

(fljj, 2/i) is

^ - Vl / x

k â€” y^ hy. â€” kx.

t.e. y = j â€” ^x+^ ^\

If this straight line touch the ellipse, it must be of the

form

y = mx + s]a^w? + h^. ( Art. 263.)

Hence

rnJ^, and f^^lJl^V = â€žW + Jl

h â€” x^ \ h â€” x^ J

Hence f ^^' " ^â– Y = Â«^ faV + 6'.

\ h-x^ J \h-xj

But this is the condition that the point (A, U) may lie

on the locus

{xy^ - x^yf =a'{y- y^f + b''{x- x^f (1).

This equation is therefore the equation to the required

tangents.

252 COORDINATE GEOMETRY.

It would be found that (1) is equivalent to

/^' ,2/' i\ /^^i' ^3// A _ f^^i ^ 2/2/1 1

279. To Jind the locus of the middle points of parallel

chords of the ellij)se.

Let the chords make with the axis an angle whose

tangent is m, so that the equation to any one of them,

QR, is

y = 7nx + c (1),

where c is different for the different chords.

This straight line meets the ellipse in points whose

abscissae are given by the equation

x^ {mx + cY

a-

-1.

i. e. ^ {a'm' + 6') + la'mcx + a'{c'-b^) = (2).

Let the roots of this equation, i.e. the abscissae of Q

and H, be x^ and x^, and let F, the middle point of QB, be

the point [h, k).

Then, by Arts. 22 and 1, we have

tJC-i *T* t^o

a^mc

C^Vl^ + IP'

(3).

CONJUGATE DIAMETERS. 253

Also V lies on the straight line (1), so that

k = mh + c (4).

If between (3) and (4) we eliminate c, we have

. a^TTb (k â€” tnh)

^" a?rri' + }f '

i. e. hVi = ~ a^mk (5).

Hence the point (A, k) always lies on the straight line

y â€” â€” 5 â€” X (6).

The required locus is therefore the straight line

y â€” 7n-,x, where m, = ^râ€” ,

^'

I.e. mm. â€” -â€ž (7).

280. Equation to the chord ivhose middle point is {h, k).

The required equation is (1) of the foregoing article, where m and

c are given by equations (4) and (5), so that

b^h , a?k^ + y-h'^

m=-^.andc = __^-.

The required equation is therefore

^~ a'k''^ a?k '

i.e. ^^{y-Jc)+~{X-h)=0.

It is therefore parallel to the polar of {h, k).

281. Diameter. Def. The locus of the middle

points of parallel chords of an ellipse is called a diameter,

and the chords are called its double ordinates.

By equation (6) of Art. 279 we see that any diameter

passes through the centre C.

Also, by equation (7), we see that the diameter 7/ â€” m^x

bisects all chords parallel to the diameter y â€” mx, if

62

mm

'-a^ W-

254 COORDINATE GEOMETRY.

But the symmetry of the result (1) shows that, in this

case, the diameter y â€” rax bisects all chords parallel to the

diameter y â€” m^x.

Such a pair of diameters are called Conjugate Diameters.

Hence

Conjugate Diameters. Def. Two diameters are

said to be conjugate when each bisects all chords parallel

to the other.

Two diameters y â€” tnx and y â€” m^x are therefore con-

jugate, if

miXli = â€” â€” g- ,

^ a-*

282. The tangent at the extremity of any diameter is

parallel to the chords which it bisects.

In the Figure of Art. 279 let [x, y') be the point P on

the ellipse, the tangent at which is parallel to the chord

QBj whose equation is

y â€” inx + c (1).

The tangent at the point (a;', y) is

^' + ^'=1 (2).

a-' o" ^ '

Since (1) and (2) are parallel, we have

a-'y

i.e. the point (cc', y^ lies on the straight line

a^m,

But, by Art. 279, this is the diameter which bisects QR

and all chords which are parallel to it.

Cor. It follows that two conjugate diameters CP and

CD are such that each is parallel to the tangent at the

extremity of the other. Hence, given either of these, we

have a geometrical construction for the other.

CONJUGATE DIAMETERS. 255

283. The tangents at the ends of any chord meet on the

diameter which bisects the chord.

Let the equation to the chord QR (Art. 279) be

2/ = ma;+ c (1).

Let T be the point of intersection of the tangents at Q

and Rj and let its coordinates be x^ and y^.

Since QR is the chord of contact of tangents from T^ its

equation is, by Art. 273,

^V^^l (2)

The equations (1) and (2) therefore represent the same

straight line, so that

})%

i.e. {h^ k) lies on the straight line

am,

which, by Art. 279, is the equation to the diameter bisect-

ing the chord QR. Hence T lies on the straight line GP.

284. If the eccentric angles of the ends, P and D, of a

pair of conjugate diameters he <^ and (f>', then <^ and cf>' differ

by a right angle.

Since P is the point [a cos ^, h sin </>), the equation to

CPis

y~x.â€” tan </> (1).

yâ€” X . - tan <^' (2).

a

So the equation to CD is

h

a

These diameters are (Art. 281) conjugate if

6^ , Â¥

â€” tan d> tan ch ~ - â€” ^ ,

a^ ^ ^ a^

i. e. if tan <^ ^^ â€” cot cf}' â€” tan (<^' Â± 90Â°j,

i.e. if ^-^'= + 90Â°.

256

COOEDINATE GEOMETRY.

Cor. 1. The points on the auxiliary circle correspond-

ing to P and D subtend a right angle at the centre.

For if p and d be these points then, by Art. 258, we

have

z.79(7^'-<^ and LdCA'^^'.

Hence

LpGd=^ LdCA' - LpGA' = <f>-<j>' = 90\

Cor. 2. In the figure of Art. 286 if P be the point <fi,

then D is the point ^ + 90Â° and D' is the point <^ - 90Â°.

285. From the previous article it follows that if P be

the point {a cos c(>, b sin <^), then D is the point

{a cos (90Â° + <^), b sin (90Â° + <^)} i- e. (â€” a sin <^, b cos </>).

Hence, if PJ^ and DM be the ordinates of P and i),

we have

iTP CM

and

CJSr MD

a

a

286. If PCP' and BCD' be a pair of conjugate dia-

meierSj then (1) CP^ + CD^ is constant, and (2) the area of

the parallelogram formed by the tangents at the ends of these

diameters is constant.

CONJUGATE DIAMETERS. 257

Let P be the point ^, so that its coordinates are a cos ^

and h sin <^. Then D is the point 90Â° + </>, so that its co-

ordinates are

a cos (90Â° + ^) and h sin (90Â°+ <^),

i.e. â€”a sin ^ and h cos <^.

(1) We therefore have

CP^ = Â«' cos^ <^ + 6' sin^ <^,

and CD'^ - a? sin^ cj> + b"^ cos^ ^,

Hence CP^ + CB' = 0"+ b'^

â€” the sum of the squares of the semi-axes of the ellipse.

(2) Let KLMN be the parallelogram formed by the

tangents at P, D, P', and D'.

By Euc. I. 36j we have

area KLMN = 4 . area CPXD

-4. GU.PK=iCU.CD,

where CU is the perpendicular from C upon the tangent

at P.

Now the equation to the tangent at P is

- cos <l> + T sin (i) â€” 1 = 0,

a

so that (Art. 75) we have

1 ab ab

C (J

/cos^

^</) sin2<^ J a" sin'' <f, + b^ cos' <f> CD

Hence CU.CI) = ab.

Thus the area of the parallelogram KLMN = iab,

which is equal to the rectangle formed by the tangents

at the ends of the major and minor axes.

287. The product of the focal distances of a point P is

equal to the square on the semidiameter parallel to the tangent

at P.

If P be the point (p, then, by Art. 251, we have

SP = a + ae cos tf), and S'P â€”a â€” ae cos ^.

L. 17

258 COORDINATE GEOMETRY.

Hence SP . S'P ^a'~ are' cos' <^

= a^ sin^ <^+b' cos^ <^

288. Ex. If P and D be the ends of conjugate diameters, find

the locus of

(1) the middle point of PD,

(2) the intersection of the tangents at P and D,

and (3) the foot of the perpendicular from the centre upon PD.

P is the point (a cos 0, h sin 0) and D is ( - a sin 0, 6 cos 0).

(1) If (x, y) be the middle point of PD, we have

a cos d>-a sin d) ^ h sin + & cos

^= ^ ^' ^^^ ^ = 2 â€¢

If we eliminate we shall get the required locus. We obtain

2 2

^ + |2=i[(cos - sin 0)2+ (sin + cos 0)2] = ^.

The locus is therefore a concentric and similar ellipse.

[N.B. Two ellipses are similar if the ratios of their axes are the

same, so that they have the same eccentricity.]

(2) The tangents are

-cos0 + vsin0=l,

a b

and - sin0 + rcos0 = l.

a b ^

Both of these equations hold at the intersection of the tangents.

If we eliminate we shall have the equation of the locus of their

intersections.

By squaring and adding, we have

so that the locus is another similar and concentric ellipse.

(3) By Art. 259, on putting 0' = 9OÂ° + 0, the equation to PD is

- cos (45Â° + 0) + 1 sin (45Â° + 0) = cos 45Â°.

Let the length of the perpendicular from the centre be 2^ and let it

make an angle w with the axis. Then this line must be equivalent to

xcosu + ysm(a=p.

CONJUGATE DIAMETERS. 259

Comparing the equations, we have

..^r. X a cos CO cos 45Â° , . , , _â€ž , . & sin w cos 45Â°

cos (45Â° + ^) = , and sin(4oÂ° + <^) =

Hence, by squaring and adding, 2p2=a2cos2 w + ft^sin^w, i.e. the

locus required is the curve

289. Equiconjugate diameters. Let P and D be ex-

tremities of equiconjugate diameters, so that CP^ = CDK

If the eccentric angle of P be 0, we then have

a^ cos^ (f> + b^ sin^ (f> = a^ sin^ (f} + h^ cos^ ^,

giving tan^ ^ = Ij

i.e. ^ = 45Â°, or 135Â°.

The equation to CP is then

h

y = x. â€” tan ^,

*.e. 2/ = Â±-^ (1)>

and that to CD is y~ â€” x â€” cot <^,

a

^.e.

2/ = + -^ (2).

If a rectangle be formed whose sides are the tangents

at Aj A', Â£, and B' the lines (1) and (2) are easily seen to

be its diagonals.

The directions of the equiconjugates are therefore along

the diagonals of the circumscribing rectangle.

The length of each equiconjugate is, by Art. 286,

290. Supplemental chords. Def. The chords

joining any point P on an ellipse to the extremities, P and

P'j of any diameter of the ellipse are called supplemental

chords.

Supplemental chords are parallel to conjugate diatneters.

17â€”2

260 COORDINATE GEOMETRY.

Let P be the point whose eccentric angle is <f>, and JR

and E' the points whose eccentric angles are <^i and

180Â° + <^i.

The equations to FB and FE' are then (Art. 259)

^ oo^ *^ + ^^ + ^ sin ilii-cos ^^1 a^

and

X ^+180Â° + ^! 2/ . ^+180Â°+^_ <^_180Â°-^

â€” cos ;r -r ^ Sm ^;; â€” â€¢ COS -z:

a 2 b 2 2 '

*.e. - - sm ^Hr^ + y cos ^ ^ ^^ = sm ^ ^ ^\ .. (2).

a 2 6 2 2 ^ '

The Â« m " of the straight line (1) = cot tÂ±il .

The " m " of the line (2) = - tan ^^^ .

CO A

If

The product of these " m's " = g , so that, by Art. 281,

the lines FR and P^' are parallel to conjugate diameters.

This proposition may also be easily proved geometrically.

For let V and V be the middle points of PJ2 and TR'.

Since V and C are respectively the middle points of EP and J2E',

the hne OF is parallel to FBI, Similarly CV is parallel to FB.

Since GV bisects PE it bisects all chords parallel to PP, i.e. all

chords parallel to GV. So CF' bisects all chords parallel to GV.

Hence CF and GV are in the direction of conjugate diameters and

therefore PP' and JPB,., being parallel to (7F and GV respectively, are

parallel to conjugate diameters.

CONJUGATE DIAMETERS. 261

291. To find the equation to an ellipse referred to a

pair of conjugate diameters.

Let the conjugate diameters be CP and CD (Fig. Art.

286), whose lengths are a' and h' respectively.

If we transform the equation to the ellipse, referred to

its principal axes, to CP and CD as axes of coordinates,

then, since the origin is unaltered, it becomes, by Art. 134,

of the form

Ax' + 2Hxy-\-By'=l (1).

Now the point P, {a, 0), lies on (1), so that

Aa" = \ (2).

So since Q, the point (0, h'), lies on (1), we have

Bh'^ = 1.

Hence A^-jz-, and B = -=-fâ€ž.

a^ h^

Also, since CP bisects all chords parallel to CD, there-

fore for each value of x we have two equal and opposite

values of y. This cannot be unless 11=0.

The equation then becomes

^+^-=1

Cor. If the axes be the equiconjugate diameters, the

equation is x^ + y^ = a'^. The equation is thus the same in

form as the equation to a circle. In the case of the ellipse

however the axes are oblique.

292. It will be noted that the equation to the ellipse,

when referred to a pair of conjugate diameters, is of the

same form as it is when referred to its principal axes.

The latter are merely a particular case of a pair of conjugate

diameters.

Just as in Art. 262, it may be shewn that the equation

to the tangent at the point (x, y') is

Similarly for the equation to the polar.

262 COORDINATE GEOMETRY.

Ex. If QVQ' be a double ordinate of the diameter CP, and if the

tangent at Q meet CP in T, then CV . CT=CP'^.

If Q be the point [x', y'), the tangent at it is

Putting y = 0, we have

^rj._a'^_GP^

I.e.

i.e. CV.CT=GP^

EXAMPLES. XXXIV.

1. In the ellipse qF + tt = 1> ^^^ ^^^ equation to the chord which

passes through the point (2, 1) and is bisected at that point.

2. Find, with respect to the elHpse 4:X^ + 7y^=8,

(1) the polar of the point ( - J, 1), and

(2) the pole of the straight line 12a; + 7t/ + 16 = 0.

3. Tangents are drawn from the point {3, 2) to the ellipse

x^ + 4:y'^=9. Find the equation to their chord of contact and the

equation of the straight line joining (3, 2) to the middle point of this

chord of contact.

4. Write down the equation of the pair of tangents drawn to the

ellipse Sx^ + 2y^=5 from the point (1, 2), and prove that the angle

between them is tan"^ â€” ^ .

a

5. In the ellipse -s + ^=l, write down the equations to the

diameters which are conjugate to the diameters whose equations are

x-y = 0, x + y = 0, y = ^x, and 2/ = -^-

6. Shew that the diameters whose equations are y + Sx = and

iy-x â€” O, are conjugate diameters of the ellipse 3x^ + 4:y^=5.

7. If the product of the perpendiculars from the foci upon the

polar of P be constant and equal to c^, prove that the locus of P is the

elHpSe 6%2 (^2 + ^2^2^ + g2^4^2^ ^4^4.

8. Shew that the four lines which join the foci to two points P

and Q on an ellipse all touch a circle whose centre is the pole of PQ.

[EXS. XXXIV.] CONJUGATE DIAMETERS. EXAMPLES. 263

9. If the pole of the normal at P lie on the normal at Q, then

shew that the pole of the normal at Q lies on the normal at P.

10. CK is the perpendicular from the centre on the polar of any

point P, and PM is the perpendicular from P on the same polar and

is produced to meet the major axis in L. Shew that (1) CK . PL = b^,

and (2) the product of the perpendiculars from the foci on the polar

= CK.LM.

What do these theorems become when P is on the ellipse ?

11. In the previous question, if PN be the ordinate of P and the

polar meet the axis in T, shew that CL = e^. CN and CT . CN-a^.

12. If tangents TP and TQ be drawn from a point T, whose

coordinates are h and k, prove that the area of the triangle TPQ is

and that the area of the quadrilateral CPTQ is

13. Tangents are drawn to the ellipse from the point

prove that they intercept on the ordinate through the nearer focus a

distance equal to the major axis.

14. Prove that the angle between the tangents that can be drawn

from any point [x^ , y{) to the ellipse is

2ah /â– ~-Â±^^yl

tan~i

x{^ + y-^ -a^-b^

15. If T be the point {x^, y-y), shew that the equation to the

straight lines joining it to the foci, S and S\ is

{x-^y - xy-yf - a\^ {y - i/i)^ = 0.

Prove that the bisector of the angle between these lines also

bisects the angle between the tangents TP and TQ that can be drawn

from T, and hence that

lSTP=lS'TQ.

16. If two tangents to an ellipse and one of its foci be given, prove

that the locus of its centre is a straight line.

17. Prove that the straight lines joining the centre to the inter-

sections of the straight line y=mx+ a/ â€” ^ with the ellipse are

conjugate diameters.

264 COOEDINATE GEOMETRY. [Exs. XXXIV.]

18. Any tangent to an ellipse meets the director circle in -p and d ;

prove that Cp and Gd are in the directions of conjugate diameters of

the ellipse.

19. If CP be conjugate to the normal at Q, prove that GQ is

conjugate to the normal at P.

20. If a fixed straight line parallel to either axis meet a pair of

conjugate diameters in the points K and L, shew that the circle

described on KL as diameter passes through two fixed points on the

other axis.

21. Prove that a chord which joins the ends of a pair of conjugate

diameters of an ellipse always touches a similar ellipse.

22. The eccentric angles of two points P and Q on the ellipse are

01 and 02 ' prove that the area of the parallelogram formed by the

tangents at the ends of the diameters through P and Q is

4a&cosec(0i-02),

and hence that it is least when P and Q are at the end of conjugate

diameters.

23. -A. pair of conjugate diameters is produced to meet the

directrix; shew that the orthocentre of the triangle so formed is at

the focus.

24. If the tangent at any point P meet in the points L and L'

(1) two parallel tangents, or (2) two conjugate diameters,

prove that in each case the rectangle LP . PL' is equal to the square

on the semidiameter which is parallel to the tangent at P.

25. -A. point is such that the perpendicular from the centre on its

polar with respect to the ellipse is constant and equal to c ; shew that

its locus is the elhpse

^2 2/2^1

26. Tangents are drawn from any point on the ellipse -3 + 'rg =1

to the circle x'^ + y'^=r'^ ; prove that the chords of contact are tangents

to the ellipse a^x^^- h^y'^ = r^.

If â€” = -5 + -s , prove that the lines joining the centre to the points

r^ a^ 0^

of contact with the circle are conjugate diameters of the second

ellipse.

27. GP and CD are conjugate diameters of the ellipse ; prove that

the locus of the orthocentre of the triangle GPD is the curve

2 {hh)'^ + aV)^^ (a2 - 62)2 (^2^2 _ ^2^2)2,

28. If circles be described on two semi-conjugate diameters of the

ellipse as diameters, prove that the locus of their second points of

intersection is the curve 2{x^-\-y'^)'^=a^x^ + h^y'^.

FOUR NORMALS TO AN ELLIPSE. 265

293. To prove that, in general, four normals can he

drawn from any point to an ellipse, and that the sum of the

eccentric angles of their feet is equal to an odd m/ultiple of

two right angles.

The normal at any point, whose eccentric angle is ^, is

r-^ = a^-h^^ de^.

cos <p sm <p

If this normal pass through the point (A, ^), we have

J!L_JL=Â«v (1).

COS </) sm <jt> ^ '

For a given point (li, k) this equation gives the

eccentric angles of the feet of the normals which pass

through {h, h).

<k

Let tan ^ = ^, so that

cos d> = , = -:, ^ , and sni cb = = .

l + tan^t lH-t-4 '^*

Substituting these values in (1), we have

ah ^ 7, â€” ok

2/,2

â€” a^e

\-f 2t

i.e. hkf + 2^3 (ah + aV) + 2i5 (ah - a^e^) -hk = ... (2).

Let ^1, ^2) hi ^^(i h ^6 the roots of this equation, so that,

by Art. 2,

^ah + a?e^ , ,

^X + ^2+^3 + ^4 = -2â€” ^^- (3),

kk + ^1^3 + ^1^4 + ^2^3 + ^2^4 + kh =^0 (4),

^ ah â€” a^p} , ,

^2^3^4 + 4^1 + Â«54^li^2 + ^1^3 = - 2 TT (5),

and ^j^2^o^4=:-l (6),

266 COOEDINATE GEOMETRY.

Hence {Trigonometry, Art. 125), we have

tan {h + ^ + il + h\ - ^1-^3 __ ^1 - ^3 _ ^

â€¢â€¢ 2 ='''^+2'

and hence <^i + ^2 + ^3 + </>4 = (2n + 1)??

= an odd multiple of two right angles.

294. We shall conclude the chapter with some ex-

amples of loci connected with the ellipse.

Ex. 1. Find the locus of the intersection of tangents at the ends

of chords of an ellipse, which are of constant length 2c.

Let QR be any such chord, and let the tangents at Q and R meet

in a point P, whose coordinates are {h, k).

Since QR is the polar of P, its equation is

The abscissae of the points in which this straight line meets the

ellipse are given by

\^ a^j ~ h^ V ^y '

L (^ fc^\ _2Â£i

x^ [W W-\ 2xh , Ti^ ^

If x^ and .^2 be the roots of this equation, i.e. the abscissae of Q

and R, we have

^l + ^2-^-2p:;-^2;i;2' and ^1^2 -j2;,2 + ^2/^2-

If 2/i and 2/2 be the ordinates of Q and R, we have from (1)

a2 "^ 62 -â€¢^'

and 2+^=1'

so that, by subtraction,

_ i'h

2/2 ~ 2/1 â€” ^2^ V.^2 ~ ^l/'

THE ELLIPSE. EXAMPLES. 267

The condition of the question therefore gives

Hence the point {h, k) always lies on the curve

^ Ka'^y^J-y &2 "^ aP ) V^ + P~-^j'

which is therefore the locus of P.

Ex. 2. Find the locus (1) of the middle points, and (2) of the poles,

of normal chords of the ellipse.

The chord, whose middle point is {h, k), is parallel to the polar of

(h, k) , and is therefore

i^-h)^,+ {y-k)^^=o (1).

If this be a normal, it must be the same as

aa^sec 6 - by cosec d = a^-h^ (2).

We therefore have

gsec 6 _ - 6 cosec 6 a^-b^

"T % ^ /i2 k-^ '

so that cos^:

a^ b^ a^ b^

and sin^=-

h{a^-b^)

fh^ Jf\

\^ '^ by '

y^'^by

k (a2 - fc2)

Hence, by the elimination of d,

fc2\2

The equation to the required locus is therefore

@4:)(^-^i=' - ^^

e4:r(^3-'-'^>=

Again, if (x^, y-y) be the pole of the normal chord (2), the latter

equation must be equivalent to the equation

^^ + ^^=1 (3).

Comparing (2) and (3), we have

a^sec0_ 6^ cosec ^

^1 vT' '

( a^ W\

so that l = cos2^ + sin2^= â€” 5+â€” 5) â€”

\x^ y^j (a

&2,

1

62)2 â€¢

268 COORDINATE GEOMETRY.

and hence the required locus is

Ex. 3. Chords of the ellipse â€” 2 + ^2 â€” â– 'â– ^^^'"^^/s touch the concentric

and coaxal ellipse -k + ^ = 1; fend the locus of their poles.

Any tangent to the second ellipse is

yz=mx+ ^Ja^m^ + p^ (1).

Let the tangents at the points where it meets the first ellipse meet

in (h, k). Then (1) must be the same as the polar of {h, k) with

respect to the first ellipse, i.e. it is the same as

a^^b'' ^~" ^''^â€¢

Since (1) and (2) coincide, we have

'h~ k~ Ti

Hence m= - ^T, and fja^m^ + jS^ == - .

a^ k

Eliminating m, we have

a4/c2 + P -/^2'

i.e. the point {h, k) lies on the ellipse

^2 1)2

i.e. on a concentric and coaxal ellipse whose semi-axes are â€” and â€”

â– a p

respectively.

EXAMPLES. XXXV.

The tangents drawn from a point P to the ellipse make angles 61

and $2 with the major axis ; find the locus of P when

1. ^1 + ^2 i^ constant (=:2a). [Compare Ex. 1, Art. 235.]

2. tan ^i + tan 62 is constant ( = c).

3. tan ^1 - tan d^ is constant ( = d!).

4. tan^ d-i + tan^ ^o is constant ( = X).

[EXS. XXXV.] THE ELLIPSE. EXAMPLES. 269

Find the locus of the intersection of tangents

5. which meet at a given angle a.

6. if the sum of the eccentric angles of their points of contact

be equal to a constant angle 2a.

7. if the difference of these eccentric angles be 120Â°.

8. if the lines joining the points of contact to the centre be

perpendicular.

9. if the sum of the ordinates of the points of contact be equal to h.

Find the locus of the midSle points of chords of an ellipse

10. whose distance from the centre is the constant length c.

11. which subtend a right angle at the centre.

12. which pass through the given point (/i, Tc).

13. whose length is constant ( = 2c).

14. whose poles are on the auxiliary circle.

15. the tangents at the ends of which intersect at right angles.

16. Prove that the locus of the intersection of normals at the

ends of conjugate diameters is the curve

2 {a?x^ + hhff= {o? - b^ {a^^^ - b^yT-

17. Prove that the locus of the intersection of normals at the ends

of chords, parallel to the tangent at the point whose eccentric angle is

a, is the conic

2 [ax sin a + by cos a) {ax cos a + by sin a) = (a^ - &2)2 g^j^ 2a cos^ 2a.

If the chords be parallel to an equiconjugate diameter, the locus

is a diameter perpendicular to the other equiconjugate.

18. A parallelogram circumscribes the ellipse and two of its

opposite angular points lie on the straight lines x'^ = h^; prove that

the locus of the other two is the conic

^2 y2

(-.:)-â–

19. Circles of constant radius c are drawn to pass through the

ends of a variable diameter of the ellipse. Prove that the locus of

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