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Hence ^ = 1 + 2 + )+-

i,e. SP.SQ.SR = SO^.a.

EXAMPLES. XXX.

Find the locus of a point O when the three normals drawn from

it are such that

1. two of them make complementary angles with the axis.

2. two of them make angles with the axis the product of whose

tangents is 2.

3. one bisects the angle between the other two.

4. two of them make equal angles with the given line y = vix + c.

5. the sum of the three angles made by them with the axis is

constant.

6. the area of the triangle formed by their feet is constant.

7. the line joining the feet of two of them is always in a given

direction.

The normals at three points P, Q, and R of the parabola y^ = 4:ax

meet in a point whose coordinates are h and k ; prove that

8. the centroid of the triangle PQR lies on the axis.

9. the point and the orthocentre of the triangle formed by the

tangents at P, Q, and R are equidistant from the axis.

[EXS. XXX.] THREE NORMALS. EXAMPLES. 215

10. if OP and OQ make complementary angles with the axis, then

the tangent at R is parallel to SO.

11. the sum of the intercepts which the normals cut off from the

axis is 2{h + a).

12. the sum of the squares of the sides of the triangle PQR is

equal to 2{h-2a){h + 10a).

13. the circle circumscribing the triangle PQR goes through the

vertex and its equation is 2x^ + 2y- -2x{h + 2a) -ky = 0.

14. if P be fixed, then QR is fixed in direction and the locus of

the centre of the circle circumscribing PQR is a straight line.

15. Three normals are drawn to the parabola y^ = 4:ax cos a from

any point lying on the straight line y = h sin a. Prove that the locus

of the orthocentre of the triangles formed by the corresponding tan-

gents is the curve -2 + t^ ~ â– 'â– ' ^^^ angle a being variable.

16. Prove that the sum of the angles which the three normals,

drawn from any point 0, make with the axis exceeds the angle which

the focal distance of O makes with the axis by a multiple of tt.

17. Two of the normals drawn from a point to the curve make

complementary angles with the axis ; prove that the locus of and

the curve which is touched by its polar are parabolas such that their

latera recta and that of the original parabola form a geometrical

progression. Sketch the three curves.

18. Prove that the normals at the points, where the straight line

lx + my = l meets the parabola, meet on the normal at the point

/4am2 4am \ â€ž , , . .

( , â€” râ€” j of the parabola.

19. If the normals at the three points P, Q, and R meet in a point

and if PP', QQ', and RR' be chords paraUel to QR, RP, and PQ

respectively, prove that the normals at P\ Q', and R' also meet in a

point.

20. If the normals drawn from any point to the parabola cut the

line x=2a in points whose ordinates are in arithmetical progres-

sion, prove that the tangents of the angles which the normals make

with the axis are in geometrical progression.

21. PG, the normal at P to a parabola, cuts the axis in G and is

produced to Q so that GQ = ^PG \ prove that the other normals

which pass through Q intersect at right angles.

22. Prove that the equation to the circle, which passes through the

focus and touches the parabola y^ = 4.ax at the point [at^, 2at), is

x^- + y^- ax {St^ + 1) - ay {St - t^) + SaH^=0.

Prove also that the locus of its centre is the curve

27 ay- = {2x - a) {x - 5a)-.

216 COORDINATE GEOMETRY. [Exs. XXX.]

23. Shew that three circles can be drawn to touch a parabola and

also to touch at the focus a given straight line passing through the

focus, and prove that the tangents at the point of contact with the

parabola form an equilateral triangle.

24. Through a point P are drawn tangents FQ and PR to a

parabola and circles are drawn through the focus to touch the para-

bola in Q and iJ respectively ; prove that the common chord of these

circles passes through the centroid of the triangle FQR.

25. Prove that the locus of the centre of the circle, which passes

through the vertex of a parabola and through its intersections with a

normal chord, is the parabola 2y'^=iax-a^.

26. -A- circle is described whose centre is the vertex and whose

diameter is three-quarters of the latus rectum of a parabola ; prove

that the common chord of the circle and parabola bisects the distance

between the vertex and the focus.

27. Prove that the sum of the angles which the four common

tangents to a parabola and a circle make with the axis is equal to

mr + 2a, where a is the angle which the radius from the focus to the

centre of the circle makes with the axis and n is an integer.

28. -P^ and QR are chords of a parabola which are normals at P

and Q. Prove that two of the common chords of the parabola and

the circle circumscribing the triangle PRQ meet on the directrix.

29. The two parabolas y^ = 4a(x-l) and x^ = 'ia(y-l') always

touch one another, the quantities I and V being both variable ; prove

that the locus of their point of contact is the curve xy='ia^.

30. A parabola, of latus rectum i^, moves so as always to touch an

equal parabola, their axes being parallel ; prove that the locus of their

point of contact is another parabola whose latus rectum is 21.

31. The sides of a triangle touch a parabola, and two of its angular

points lie on another parabola with its axis in the same direction ;

prove that the locus of the third angular point is another parabola.

239. In Art. 197 we obtained the simplest possible

form of the equation to a parabola.

We shall now transform the origin and axes in the

most general manner.

Let the new origin have as coordinates [h, k), and let

the new axis of x be inclined at to the original axis, and

let the new angle between the axes be o>'.

PAEABOLA. TWO TANGENTS AS AXES. 217

By Art. 133 we have for x and y to substitute

X cos B -^y cos (w' + ^) + A,

and X sin Q + y sin (to' + ^) + ^

respectively.

The equation of Art. 197 then becomes

{x sin B-\-y sin (o>' + ^) + ^|^ = 4a {x cos ^ + y cos (w' + 6^) + A},

i.e.

\x sin ^ + 2/ sin (<o' + ^)}^ + 2Â£c {A; sin ^ - 2a cos ^}

+ 1y {h sin (w' + ^) - 2a cos (w' + ^)} + ^^^ _ 4^;^ ^ q

â– â€¢â€¢â€¢â€¢â€¢â€¢(l)-

This equation is therefore the most general form of the

equation to a parabola.

We notice that in it the terms of the second degree

always form a perfect square.

240. To find the equation to a 'parabola, any two

tangents to it being the axes of coordinates and the points of

contact being distant a and b from the origin.

By the last article the most general form of the equa-

tion to any parabola is

{Ax-^Byf-v2gx+2fy-^c = (1).

This meets the axis of x in points whose abscissae are

given by

^ V + ^gx + c = (2).

If the parabola touch the axis of x at a distance a from

the origin, this equation must be equivalent to

A^{x-af = (3).

Comparing equations (2) and (3), we have

g- â€” A^a, and c = A^a^ (4).

Similarly, since the parabola is to touch the axis of y

at a distance b from the origin, we have

f=-B%, and c=^B''b' (5).

218 COORDINATE GEOMETRY.

From (4) and (5), equating the values of c, we have

sothat B = Â±A^ (6).

Taking the negative sign, we have

B = -A'^, g = -'A\ f^-A'j, and G=A'a\

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

equation,

t.e.

('?_iy_?^_^+i=o (7).

\a 0/ a

This equation can be written in the form

\a 0/ \a 0/ ao

ah'

yivi=i <Â«)â€¢

a

I.e. I ^ / -+ /v/ ^j =lj

^.e.

[The radical signs in (8) can clearly have both the positive and

negative signs prefixed. The different equations thus obtained corre-

spond to different portions of the curve. In the figure of Art. 243,

the abscissa of any point on the portion PAQ is <a, and the ordinate

< h, so that for this portion of the curve we must take both signs

positive. For the part beyond P the abscissa is > a, and - > r , so

that the signs must be + and - . For the part beyond Q the

fit n^

ordinate is >&, and v>-, so that the signs must be - and +.

a

There is clearly no part of the cui-ve corresponding to two negative

signs.]

PARABOLA. TWO TANGENTS AS AXES. 219

241. If in the previous article we took the positive

sign in (6), the equation would reduce to

(- + I-) _2 - ^ + l = 0,

\a J a

This gives us (Fig., Art. 243) the pair of coincident

straight lines PQ. This pair of coincident straight lines is

also a conic meeting the axes in two coincident points at P

and Q, but is not the parabola required.

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

(x', y') of the parabola

Let (cc", 2/") be any point on the curve close to {x ^ y).

The equation to the line joining these two points is

y - 2/' = |iJ~' (Â«-Â«') (1)-

But, since these points lie on the curve, we have

Ix

sl ~a

^ . /y 1 M' , Ivi

+ ^/| = l = Vl^-^/y (2).

so that ^gz^ = -^ (3).

sjx' â€” s]x s]a

The equation (1) is therefore

/ _ sly" - -Jy sly" + ^ y , _ ,.

y y ~ Iâ€” i~, j-r, , r,\^ ^n

sJX â€” \JX \JX + sjX !

or, by (3),

; , sjh sly" + sjy , ,. ...

y-y= - ,- -h, â€” F=X^-^) (4).

sja six + ycc

220 COORDINATE GEOMETRY.

The equation to the tangent at {x\ y') is then obtained

by putting x" = x and y" â€” y', and is

Jh sly'

y-y = - 7- -7^ (x-x),

s/a \Jx

%.e.

slax''^ slhy'~ \' a ^ \ h~^ ^^^*

This is the required equation.

[In the foregoing we have assumed that [x', y') Hes on the portion

PAQ (Fig., Art. 243). If it lie on either of the other portions the

proper signs must be affixed to the radicals, as in Art. 240.]

CO u

Ex. To find the condition that the straight line ^+ - = lmay be a

tangent.

This line will be the same as (5), if

f = Jax' and g=Jhy',

1^' f fv' Q

SO that s. â€”~-i and m.I^ â€” t-

'V '' Â« \' h

Hence - + ^ = 1.

a

This is the required condition; also, since x'=â€” and ^'=â€” ,

the point of contact of the given line is ( "â€” , ^ J .

Similarly, the straight line lx + my = n will touch the parabola if

n ** _i

al bm '

243. To find the focus of the j^arabola

a W h

Let ^S' be the focus, the origin, and P and Q the

points of contact of the parabola with the axes.

Since, by Art. 230, the triangles OSP and QSO are

similar, the angle SOP- angle SQO.

Hence if we describe a circle through 0, Q, and S, then,

by Euc. III. 32, OP is the tangent to it at P.

PARABOLA. TWO TANGENTS AS AXES. 221

Hence S lies on the circle passing through the origin

0, the point Q, (0, b), and touching the axis of x at the

origin.

P X

The equation to this circle is

x^ + 2xy cos oi +y^ = by ( 1 ).

Similarly, since z SOQ â€” L SPO^ S will lie on the circle

through and P and touching the axis of y at the origin,

i.e. on the circle

x^ + 2xy cos (0 +y'^ â€” ax (2).

The intersections of (1) and (2) give the point required.

On solving (1) and (2), we have as the focus the point

air" a^h

qP' + 2ah cos <o + 6" ' ffl^ + 2ah cos <o + 6'

244. To find the equation to the axis.

If V be the middle point of PQ^ we know, by Art. 223,

that F is parallel to the axis.

Now V is the point ( - , - ) .

Hence the equation to F is y = -x.

The equation to the axis (a line through S parallel to

F) is therefore

a^h h ( ah^ \

If r= â€” I Â£C â€” â– I

a^ + 2ab cos w +b^ a\ a^ + 2ab cos w + 6v '

, ab (a? â€” b^)

*â€¢ ^- ay-bx= ~ â€” â€”\ ^ â€” - .

a^ + lab cos o) + 6"

222 COORDINATE GEOMETRY.

245 . To find the equation to the directrix.

If we find the point of intersection of OP and a

tangent perpendicular to OP, this point will (Art. 211, y)

be on the directrix.

Similarly we can obtain the point on OQ which is on

the directrix.

A straight line through the point (/, 0) perpendicular

to OX is

y = in{x â€”f), where (Art. 93) 1 +7n cos o> = 0.

The equation to this perpendicular straight line is

then

x+ 2/ cos (0 =y (1).

This straight line touches the parabola if (Art. 242)

/ / 1 â€¢ -I. ^' Â«^ cos

(X)

a b cos o) ' ' a+ b cos to

ab cos (0

The point ( â€” â€” = , ) therefore lies on the directrix.

\a + b cos <o /

^. ., , 1 . //^ ab cos, (0 \ .

Similarly the point ( 0, , ) is on it.

â€¢^ ^ \ b +a cos (0/

The equation to the directrix is therefore

x{a + b cos 0)) +1/ (b + a cos w) = ab cos oo (2).

The latus rectum being twice the perpendicular distance

of the focus from the directrix â€” twice the distance of the

point

/ ab^ a'^b \

\aP + 2ab cos oo + 5^ ' a"^ + 2ab cos w + b^J

from the straight line (2)

4<x^6^ sin^ w

~ (a^ + 2ab cos oo + 6^)t '

by Art. 96, after some reduction.

246. To find the coordinates of the vertex aiid the

equation to the tangent at the vertex.

PARABOLA. TWO TANGENTS AS AXES.

223

The vertex is the intersection of the axis and the curve,

i.e. its coordinates are given by

H X a? â€” h^ .

and by

i.e. by

yy

h

a? + 2ah cos <o + 6^

X

fx

\a

-^-1+1 =

a

From (1) a,nd (2), we have

x =

1

O 7 <>

a" â€” 0-

a?' + lab cos a> + 6^

alP" {h â– \- a cos w)^

.(Art. 240),

(2).

{ci? + 2a& cos (0 + IP'Y '

Similarly y

a^b (a + b cos tof

(a? + 2ab cos w + b^Y '

These are the coordinates of the vertex.

The tangent at the vertex being parallel to the directrix,

its equation is

Â«6^ {b + a cos uif

(a + b cos co)

X â€”

a^ + 2ab cos oo + b^f

,-, . r a%(a + b cos o>)"^ H .

+ (6 + Â« cos co) 2/ - jâ€”^ )r-j~ -^ r = 0,

^ L (Â« + 2ao cos (o + Â¥fj

^.e.

â€” +

2/

Â«.6

b + a cos (0 a + 5 cos co a^ + 2ab cos co + 6^

EXAMPLES. XXXI.

1. If a parabola, whose latus rectum is 4c, slide between two

rectangular axes, prove that the locus of its focus is x'^y^=c^ {x^ + y^),

and that the curve traced out by its vertex is

2. Parabolas are drawn to touch two given rectangular axes and

their foci are all at a constant distance c from the origin. Prove that

the locus of the vertices of these parabolas is the curve

x^ + y^=c'"

224 COOKDINATE GEOMETRY. [ExS. XXXI.]

3. The axes being rectangular, prove that the locus of the focus

of the parabola (- + r-l) = â€” t j ^ 3,nd h being variables such

\a j ah

that db = c^, is the curve {x'^-\-']ff â€” e^xy.

4. Parabolas are drawn to touch two given straight lines which

are inclined at an angle w ; if the chords of contact all pass through

a fixed point, prove that

(1) their directrices all pass through another fixed point, and

(2) their foci all lie on a circle which goes through the intersection of

the two given straight lines.

5. A parabola touches two given straight lines at given points ;

prove that the locus of the middle point of the portion of any tangent

which is intercepted between the given straight lines is a straight

line.

6. TP and TQ are any two tangents to a parabola and the

tangent at a third point B, cuts them in F' and Q' ; prove that

TT' Tq_ QQ' _TP' _Q'R

7. If a parabola touch three given straight lines, prove that each

of the lines joining the points of contact passes through a fixed point.

8. A parabola touches two given straight lines ; if its axis pass

through the point {h, k), the given lines being the axes of coordinates,

prove that the locus of the focus is the curve

x'^-y^-hx + ky=: 0.

9. A parabola touches two given straight lines, which meet at O,

in given points and a variable tangent meets the given lines in P and

Q respectively ; prove that the locus of the centre of the circumcircle

of the triangle OPQ is a fixed straight line.

10. The sides AB and AC of a triangle ABC are given in position

and the harmonic mean between the lengths AB a,nd AC is also given;

prove that the locus of the focus of the parabola touching the sides at

B and C is a circle whose centre lies on the line bisecting the angle

BAG.

11. Parabolas are drawn to touch the axes, which are inclined at

an angle w, and their directrices all pass through a fixed point (h, k).

Prove that all the parabolas touch the straight line

+ ^r-r^ = 1-

h + k sec 0} k + h sec w

CHAPTER XII.

THE ELLIPSE.

247. The ellipse is a conic section in which the

eccentricity e is less than unity.

To find the equation to an ellipse.

Let ZK be the directrix, S the focus, and let SZ be

perpendicular to the directrix.

There will be a point A on SZ, such that

SA = e.AZ (1).

Since e < 1, there will be another point A\ on ZS produced,

such that

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

L. 15

226 COORDINATE GEOMETRY.

Let the length A A! be called 2<a^, and let C be the middle

point of AA', Adding (1) and (2), we have

'la=-AA! = e{AZ^A:Z) = 'l.e.CZ,

i.e. CZ = - (3).

Subtracting (1) from (2), we have

e{A:Z-AZ)^SA'-8A^{SG^CA')-{GA-CS),

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

and hence CS â€” a.e (4).

Let G be the origin, CA' the axis of x, and a line through

C perpendicular to ^^' the axis of 3/.

Let P be any point on the curve, whose coordinates are

X and y, and let Pif be the perpendicular upon the directrix,

and PJV the perpendicular upon A A'.

The focus S is the point (â€” ae, 0).

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

(x + aef-Â¥y^^e^(x+'^\ (Art. 20),

i.e x%l^e') + y' = a\l-e%

a'{l-e')

If in this equation we put x = Oy we have

y = + a\/lâ€”e^j

shewing that the curve meets the axis of y in two points,

B and B', lying on opposite sides of C, such that

B'C ^CB-^a sjT^\ i.e. CB' = CA' - CS\

Let the length CB be called 6, so that

The equation (5) then becomes

Â±-+L = l (6).

a2^b2"-^ ^ ^

i-e. -2 + -2^^i â€” :i^ = l (^)-

THE ELLIPSE. 227

248. The equation (6) of the previous article may be

written

if' Q(? c? â€” x^ {a + x) {a â€” x)

Jf a^ c^ o? '

FN'' AN.NA'

i.e. PJV^ : AN.NA' :: BC : AC\

Def. The points A and A' are called the vertices of

the curve, AA' is called the major axis, and BB' the minor

axis. Also C is called the centre.

249. Since â™¦S' is the point {â€”ae, 0), the equation to

the ellipse referred to S as origin is (Art. 128),

The equation referred to A as origin, and AX and a

perpendicular line as axes, is

(^-^)' , ^' _ 1

a- 0^ a

Similarly, the equation referred to ZX and ZK as axes is,

a

since CZ= â€” ,

(-3'

+ ^- = h

The equation to the ellipse, whose focus and directrix are any

given point and line, and whose eccentricity is known, is easily

written down.

For example, if the focus be the point (-2, 3), the directrix be

the line 2a; + 3?/ + 4 = 0, and the eccentricity be |, the required equa-

tion is

i.e. 261*2 + 181i/2 - 192a;?/ + 1044a; - 2334?/ + 3969 = 0.

15â€”2

228 COORDINATE GEOMETRY.

Generally, the equation to the ellipse, whose focus is the point

(/> P)> whose directrix is Ax + By + C = 0, and whose eccentricity

is e, is

250. There exist a second focus and a second directrix

for the curve.

On the positive side of the origin take a point S', which

is such that SO â€” CS' = ae, and another point Z', such that

e

Draw Z'K' perpendicular to ZZ\ and PM' perpen-

dicular to Z'K'.

The equation (5) of Art. 247 may be written in the

form

x^ â€” 2aex + ah^ + y^ = ^'^^ - 2aeaj + c?,

i.e. {x - aef -^ y^ = e^ (- - x\ ,

i.e. ST^^e^PM"".

Hence any point P of the curve is such that its distance

from S' is e times its distance from Z'K', so that we should

have obtained the same curve, if we had started with S' as

focus, Z'K' as directrix, and the same eccentricity.

251. The sum of the focal distances of any point on the

curve is equal to the major axis.

For (Fig. Art. 247) we have

SP = e.PM, and S'P^e.PM'.

Hence

SP + S'P = e (PM+PM') = e . MM'

= e.ZZ' = 2e.CZ=^2a (Art. 247.)

= the major axis.

Also SP - e . PM^ e.NZ^e.CZ+e.CN-?i + ex',

and S'P - e . PM' = e . NZ' = e . CZ' - e . CN -?i- ex',

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

THE ELLIPSE. LATUS-RECTUM. 229

252. Mechanical construction for an ellipse.

By the preceding article we can get a simple mechanical

method of constructing an ellipse.

Take a piece of thread, whose length is the major axis

of the required ellipse, and fasten its ends at the points S

and aS" which are to be the foci.

Let the point of a pencil move on the paper, the point

being always in contact with the string and keeping the

two portions of the string between it and the fixed ends

always tight. If the end of the pencil be moved about on

the paper, so as to satisfy these conditions, it will trace out

the curve on the paper. For the end of the pencil will be

always in such a position that the sum of its distances from

S and S' will be constant.

In practice, it is easier to fasten two drawing pins at S

and aS", and to have an endless piece of string whose total

length is equal to the sum of SS' and AA'. This string

must be passed round the two pins at S and aS" and then be

kept stretched by the pencil as before. By this second

arrangement it will be found that the portions of the curve

near A and A' can be more easily described than in the first

method.

253. Latus-rectum of the ellipse.

Let LSL' be the double ordinate of the curve which

passes through the focus aS'. By the definition of the curve,

the semi-latus-rectum SL

= e times the distance of L from the directrix

^e.SZ=e{CZ-CS) = e.CZ-e.CS

= a â€” ae^ (by equations (3) and (4) of Art. 247)

= -. (Art. 247.)

254. To trace the curve

%4-' Â«â€¢

230 COORDINATE GEOMETRr.

The equation may be written in either of the forms

Â±Â«yi-s (3).

or X

From (2), it follows that if Q^>a?, i.e. if x> a ov <- a^

then y is impossible. There is therefore no part of the

curve to the right of J.' or to the left of A.

From (3), it follows, similarly, that, if y>h or <:-6,

X is impossible, and hence that there is no part of the curve

above B or below B'.

If X lie between â€” a and + a^ the equation (2) gives two

equal and opposite values for y^ so that the curve is sym-

metrical with respect to the axis of x.

If y lie between â€” h and + 5, the equation (3) gives two

equal and opposite values for x^ so that the curve is sym-

metrical with respect to the axis of y.

If a number of values in succession be given to Â£c, and

the corresponding values of y be determined, we shall

obtain a series of points which will all be found to lie on a

curve of the shape given in the figure of Art. 247.

255. The quantity â€” ^ + -7^ â€” 1 i^ negative^ zero, or

â‚¬(/

positive, according as the point (x , y') lies vjithin, upon, or

without the ellijjse.

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

tlirough Q meet the curve in P, so that, by equation (6) of

Art. 247,

PiP _ 1 ^'

"W ~ ^ â€¢

If Q be within the curve, then y', i.e. QI^, is < FN, so

that

V'' PN^ . -, x'^

0- h- a^

RADIUS VECTOR IN ANY DIRECTION. 231

Hence, in this case,

I.e. â€” r + "v^â€” 1 IS negative.

Similarly, if Q' be without the curve, y' > PN, and then

'2 '2

-^ + ^ - 1 is positive,

a" 6^

256. To find the length of a radius vector from the

centre drawn in a given direction.

The equation (6) of Art. 247 when transferred to polar

coordinates becomes

r^cos^^ r^sin^^

a'W

We thus have the value of the radius vector drawn at any

inclination 6 to the axis.

Since r^ = ^a â€” r-s â€” tit^ â€” r-â€ž-;, , we see that the greatest

62 + (a2-62)sin2(9' *=

value of r is when ^ = 0, and then it is equal to a.

Similarly, B == 90Â° gives the least value of r, viz. h.

Also, for each value of ^, we have two equal and opposite

values of r, so that any line through the centre meets the

curve in two points equidistant from it.

257. Auxiliary circle. Def. The circle which is

described on the major axis, AA\ of an ellipse as diameter,

is called the auxiliary circle of the ellipse.

Let NP be any ordinate of the ellipse, and let it be

produced to meet the auxiliary circle in Q.

Since the angle AQA' is a right angle, being the angle

in a semicircle, we have, by Euc. vi. 8, QN^ = AN. NA'.

232

COORDINATE GEOMETRY.

SO tliat

Hence Art. 248 gives

PN^ : QN' :: BC^ : AG\

PNBCb

QN""AC"a'

Y

^^^^

Q'

y^^^-^^

P

"^^"^^^

f .

P

t^-...

\ C N' N jA' X 1

The point Q in which the ordinate NF meets the

auxiliary circle is called the corresponding point to P.

The ordinates of any point on the ellipse and the

corresponding point on the auxiliary circle are therefore to

one another in the ratio h : Â«, i.e. in the ratio of the

semi-minor to the semi-major axis of the ellipse.

The ellipse might therefore have been defined as follows :

Take a circle and from each point of it draw perpen-

diculars upon a diameter ; the locus of the points dividing

these perpendiculars in a given ratio is an ellipse, of which

the given circle is the auxiliary circle.

258. Eccentric Angle. Def. The eccentric angle

i,e. SP.SQ.SR = SO^.a.

EXAMPLES. XXX.

Find the locus of a point O when the three normals drawn from

it are such that

1. two of them make complementary angles with the axis.

2. two of them make angles with the axis the product of whose

tangents is 2.

3. one bisects the angle between the other two.

4. two of them make equal angles with the given line y = vix + c.

5. the sum of the three angles made by them with the axis is

constant.

6. the area of the triangle formed by their feet is constant.

7. the line joining the feet of two of them is always in a given

direction.

The normals at three points P, Q, and R of the parabola y^ = 4:ax

meet in a point whose coordinates are h and k ; prove that

8. the centroid of the triangle PQR lies on the axis.

9. the point and the orthocentre of the triangle formed by the

tangents at P, Q, and R are equidistant from the axis.

[EXS. XXX.] THREE NORMALS. EXAMPLES. 215

10. if OP and OQ make complementary angles with the axis, then

the tangent at R is parallel to SO.

11. the sum of the intercepts which the normals cut off from the

axis is 2{h + a).

12. the sum of the squares of the sides of the triangle PQR is

equal to 2{h-2a){h + 10a).

13. the circle circumscribing the triangle PQR goes through the

vertex and its equation is 2x^ + 2y- -2x{h + 2a) -ky = 0.

14. if P be fixed, then QR is fixed in direction and the locus of

the centre of the circle circumscribing PQR is a straight line.

15. Three normals are drawn to the parabola y^ = 4:ax cos a from

any point lying on the straight line y = h sin a. Prove that the locus

of the orthocentre of the triangles formed by the corresponding tan-

gents is the curve -2 + t^ ~ â– 'â– ' ^^^ angle a being variable.

16. Prove that the sum of the angles which the three normals,

drawn from any point 0, make with the axis exceeds the angle which

the focal distance of O makes with the axis by a multiple of tt.

17. Two of the normals drawn from a point to the curve make

complementary angles with the axis ; prove that the locus of and

the curve which is touched by its polar are parabolas such that their

latera recta and that of the original parabola form a geometrical

progression. Sketch the three curves.

18. Prove that the normals at the points, where the straight line

lx + my = l meets the parabola, meet on the normal at the point

/4am2 4am \ â€ž , , . .

( , â€” râ€” j of the parabola.

19. If the normals at the three points P, Q, and R meet in a point

and if PP', QQ', and RR' be chords paraUel to QR, RP, and PQ

respectively, prove that the normals at P\ Q', and R' also meet in a

point.

20. If the normals drawn from any point to the parabola cut the

line x=2a in points whose ordinates are in arithmetical progres-

sion, prove that the tangents of the angles which the normals make

with the axis are in geometrical progression.

21. PG, the normal at P to a parabola, cuts the axis in G and is

produced to Q so that GQ = ^PG \ prove that the other normals

which pass through Q intersect at right angles.

22. Prove that the equation to the circle, which passes through the

focus and touches the parabola y^ = 4.ax at the point [at^, 2at), is

x^- + y^- ax {St^ + 1) - ay {St - t^) + SaH^=0.

Prove also that the locus of its centre is the curve

27 ay- = {2x - a) {x - 5a)-.

216 COORDINATE GEOMETRY. [Exs. XXX.]

23. Shew that three circles can be drawn to touch a parabola and

also to touch at the focus a given straight line passing through the

focus, and prove that the tangents at the point of contact with the

parabola form an equilateral triangle.

24. Through a point P are drawn tangents FQ and PR to a

parabola and circles are drawn through the focus to touch the para-

bola in Q and iJ respectively ; prove that the common chord of these

circles passes through the centroid of the triangle FQR.

25. Prove that the locus of the centre of the circle, which passes

through the vertex of a parabola and through its intersections with a

normal chord, is the parabola 2y'^=iax-a^.

26. -A- circle is described whose centre is the vertex and whose

diameter is three-quarters of the latus rectum of a parabola ; prove

that the common chord of the circle and parabola bisects the distance

between the vertex and the focus.

27. Prove that the sum of the angles which the four common

tangents to a parabola and a circle make with the axis is equal to

mr + 2a, where a is the angle which the radius from the focus to the

centre of the circle makes with the axis and n is an integer.

28. -P^ and QR are chords of a parabola which are normals at P

and Q. Prove that two of the common chords of the parabola and

the circle circumscribing the triangle PRQ meet on the directrix.

29. The two parabolas y^ = 4a(x-l) and x^ = 'ia(y-l') always

touch one another, the quantities I and V being both variable ; prove

that the locus of their point of contact is the curve xy='ia^.

30. A parabola, of latus rectum i^, moves so as always to touch an

equal parabola, their axes being parallel ; prove that the locus of their

point of contact is another parabola whose latus rectum is 21.

31. The sides of a triangle touch a parabola, and two of its angular

points lie on another parabola with its axis in the same direction ;

prove that the locus of the third angular point is another parabola.

239. In Art. 197 we obtained the simplest possible

form of the equation to a parabola.

We shall now transform the origin and axes in the

most general manner.

Let the new origin have as coordinates [h, k), and let

the new axis of x be inclined at to the original axis, and

let the new angle between the axes be o>'.

PAEABOLA. TWO TANGENTS AS AXES. 217

By Art. 133 we have for x and y to substitute

X cos B -^y cos (w' + ^) + A,

and X sin Q + y sin (to' + ^) + ^

respectively.

The equation of Art. 197 then becomes

{x sin B-\-y sin (o>' + ^) + ^|^ = 4a {x cos ^ + y cos (w' + 6^) + A},

i.e.

\x sin ^ + 2/ sin (<o' + ^)}^ + 2Â£c {A; sin ^ - 2a cos ^}

+ 1y {h sin (w' + ^) - 2a cos (w' + ^)} + ^^^ _ 4^;^ ^ q

â– â€¢â€¢â€¢â€¢â€¢â€¢(l)-

This equation is therefore the most general form of the

equation to a parabola.

We notice that in it the terms of the second degree

always form a perfect square.

240. To find the equation to a 'parabola, any two

tangents to it being the axes of coordinates and the points of

contact being distant a and b from the origin.

By the last article the most general form of the equa-

tion to any parabola is

{Ax-^Byf-v2gx+2fy-^c = (1).

This meets the axis of x in points whose abscissae are

given by

^ V + ^gx + c = (2).

If the parabola touch the axis of x at a distance a from

the origin, this equation must be equivalent to

A^{x-af = (3).

Comparing equations (2) and (3), we have

g- â€” A^a, and c = A^a^ (4).

Similarly, since the parabola is to touch the axis of y

at a distance b from the origin, we have

f=-B%, and c=^B''b' (5).

218 COORDINATE GEOMETRY.

From (4) and (5), equating the values of c, we have

sothat B = Â±A^ (6).

Taking the negative sign, we have

B = -A'^, g = -'A\ f^-A'j, and G=A'a\

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

equation,

t.e.

('?_iy_?^_^+i=o (7).

\a 0/ a

This equation can be written in the form

\a 0/ \a 0/ ao

ah'

yivi=i <Â«)â€¢

a

I.e. I ^ / -+ /v/ ^j =lj

^.e.

[The radical signs in (8) can clearly have both the positive and

negative signs prefixed. The different equations thus obtained corre-

spond to different portions of the curve. In the figure of Art. 243,

the abscissa of any point on the portion PAQ is <a, and the ordinate

< h, so that for this portion of the curve we must take both signs

positive. For the part beyond P the abscissa is > a, and - > r , so

that the signs must be + and - . For the part beyond Q the

fit n^

ordinate is >&, and v>-, so that the signs must be - and +.

a

There is clearly no part of the cui-ve corresponding to two negative

signs.]

PARABOLA. TWO TANGENTS AS AXES. 219

241. If in the previous article we took the positive

sign in (6), the equation would reduce to

(- + I-) _2 - ^ + l = 0,

\a J a

This gives us (Fig., Art. 243) the pair of coincident

straight lines PQ. This pair of coincident straight lines is

also a conic meeting the axes in two coincident points at P

and Q, but is not the parabola required.

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

(x', y') of the parabola

Let (cc", 2/") be any point on the curve close to {x ^ y).

The equation to the line joining these two points is

y - 2/' = |iJ~' (Â«-Â«') (1)-

But, since these points lie on the curve, we have

Ix

sl ~a

^ . /y 1 M' , Ivi

+ ^/| = l = Vl^-^/y (2).

so that ^gz^ = -^ (3).

sjx' â€” s]x s]a

The equation (1) is therefore

/ _ sly" - -Jy sly" + ^ y , _ ,.

y y ~ Iâ€” i~, j-r, , r,\^ ^n

sJX â€” \JX \JX + sjX !

or, by (3),

; , sjh sly" + sjy , ,. ...

y-y= - ,- -h, â€” F=X^-^) (4).

sja six + ycc

220 COORDINATE GEOMETRY.

The equation to the tangent at {x\ y') is then obtained

by putting x" = x and y" â€” y', and is

Jh sly'

y-y = - 7- -7^ (x-x),

s/a \Jx

%.e.

slax''^ slhy'~ \' a ^ \ h~^ ^^^*

This is the required equation.

[In the foregoing we have assumed that [x', y') Hes on the portion

PAQ (Fig., Art. 243). If it lie on either of the other portions the

proper signs must be affixed to the radicals, as in Art. 240.]

CO u

Ex. To find the condition that the straight line ^+ - = lmay be a

tangent.

This line will be the same as (5), if

f = Jax' and g=Jhy',

1^' f fv' Q

SO that s. â€”~-i and m.I^ â€” t-

'V '' Â« \' h

Hence - + ^ = 1.

a

This is the required condition; also, since x'=â€” and ^'=â€” ,

the point of contact of the given line is ( "â€” , ^ J .

Similarly, the straight line lx + my = n will touch the parabola if

n ** _i

al bm '

243. To find the focus of the j^arabola

a W h

Let ^S' be the focus, the origin, and P and Q the

points of contact of the parabola with the axes.

Since, by Art. 230, the triangles OSP and QSO are

similar, the angle SOP- angle SQO.

Hence if we describe a circle through 0, Q, and S, then,

by Euc. III. 32, OP is the tangent to it at P.

PARABOLA. TWO TANGENTS AS AXES. 221

Hence S lies on the circle passing through the origin

0, the point Q, (0, b), and touching the axis of x at the

origin.

P X

The equation to this circle is

x^ + 2xy cos oi +y^ = by ( 1 ).

Similarly, since z SOQ â€” L SPO^ S will lie on the circle

through and P and touching the axis of y at the origin,

i.e. on the circle

x^ + 2xy cos (0 +y'^ â€” ax (2).

The intersections of (1) and (2) give the point required.

On solving (1) and (2), we have as the focus the point

air" a^h

qP' + 2ah cos <o + 6" ' ffl^ + 2ah cos <o + 6'

244. To find the equation to the axis.

If V be the middle point of PQ^ we know, by Art. 223,

that F is parallel to the axis.

Now V is the point ( - , - ) .

Hence the equation to F is y = -x.

The equation to the axis (a line through S parallel to

F) is therefore

a^h h ( ah^ \

If r= â€” I Â£C â€” â– I

a^ + 2ab cos w +b^ a\ a^ + 2ab cos w + 6v '

, ab (a? â€” b^)

*â€¢ ^- ay-bx= ~ â€” â€”\ ^ â€” - .

a^ + lab cos o) + 6"

222 COORDINATE GEOMETRY.

245 . To find the equation to the directrix.

If we find the point of intersection of OP and a

tangent perpendicular to OP, this point will (Art. 211, y)

be on the directrix.

Similarly we can obtain the point on OQ which is on

the directrix.

A straight line through the point (/, 0) perpendicular

to OX is

y = in{x â€”f), where (Art. 93) 1 +7n cos o> = 0.

The equation to this perpendicular straight line is

then

x+ 2/ cos (0 =y (1).

This straight line touches the parabola if (Art. 242)

/ / 1 â€¢ -I. ^' Â«^ cos

(X)

a b cos o) ' ' a+ b cos to

ab cos (0

The point ( â€” â€” = , ) therefore lies on the directrix.

\a + b cos <o /

^. ., , 1 . //^ ab cos, (0 \ .

Similarly the point ( 0, , ) is on it.

â€¢^ ^ \ b +a cos (0/

The equation to the directrix is therefore

x{a + b cos 0)) +1/ (b + a cos w) = ab cos oo (2).

The latus rectum being twice the perpendicular distance

of the focus from the directrix â€” twice the distance of the

point

/ ab^ a'^b \

\aP + 2ab cos oo + 5^ ' a"^ + 2ab cos w + b^J

from the straight line (2)

4<x^6^ sin^ w

~ (a^ + 2ab cos oo + 6^)t '

by Art. 96, after some reduction.

246. To find the coordinates of the vertex aiid the

equation to the tangent at the vertex.

PARABOLA. TWO TANGENTS AS AXES.

223

The vertex is the intersection of the axis and the curve,

i.e. its coordinates are given by

H X a? â€” h^ .

and by

i.e. by

yy

h

a? + 2ah cos <o + 6^

X

fx

\a

-^-1+1 =

a

From (1) a,nd (2), we have

x =

1

O 7 <>

a" â€” 0-

a?' + lab cos a> + 6^

alP" {h â– \- a cos w)^

.(Art. 240),

(2).

{ci? + 2a& cos (0 + IP'Y '

Similarly y

a^b (a + b cos tof

(a? + 2ab cos w + b^Y '

These are the coordinates of the vertex.

The tangent at the vertex being parallel to the directrix,

its equation is

Â«6^ {b + a cos uif

(a + b cos co)

X â€”

a^ + 2ab cos oo + b^f

,-, . r a%(a + b cos o>)"^ H .

+ (6 + Â« cos co) 2/ - jâ€”^ )r-j~ -^ r = 0,

^ L (Â« + 2ao cos (o + Â¥fj

^.e.

â€” +

2/

Â«.6

b + a cos (0 a + 5 cos co a^ + 2ab cos co + 6^

EXAMPLES. XXXI.

1. If a parabola, whose latus rectum is 4c, slide between two

rectangular axes, prove that the locus of its focus is x'^y^=c^ {x^ + y^),

and that the curve traced out by its vertex is

2. Parabolas are drawn to touch two given rectangular axes and

their foci are all at a constant distance c from the origin. Prove that

the locus of the vertices of these parabolas is the curve

x^ + y^=c'"

224 COOKDINATE GEOMETRY. [ExS. XXXI.]

3. The axes being rectangular, prove that the locus of the focus

of the parabola (- + r-l) = â€” t j ^ 3,nd h being variables such

\a j ah

that db = c^, is the curve {x'^-\-']ff â€” e^xy.

4. Parabolas are drawn to touch two given straight lines which

are inclined at an angle w ; if the chords of contact all pass through

a fixed point, prove that

(1) their directrices all pass through another fixed point, and

(2) their foci all lie on a circle which goes through the intersection of

the two given straight lines.

5. A parabola touches two given straight lines at given points ;

prove that the locus of the middle point of the portion of any tangent

which is intercepted between the given straight lines is a straight

line.

6. TP and TQ are any two tangents to a parabola and the

tangent at a third point B, cuts them in F' and Q' ; prove that

TT' Tq_ QQ' _TP' _Q'R

7. If a parabola touch three given straight lines, prove that each

of the lines joining the points of contact passes through a fixed point.

8. A parabola touches two given straight lines ; if its axis pass

through the point {h, k), the given lines being the axes of coordinates,

prove that the locus of the focus is the curve

x'^-y^-hx + ky=: 0.

9. A parabola touches two given straight lines, which meet at O,

in given points and a variable tangent meets the given lines in P and

Q respectively ; prove that the locus of the centre of the circumcircle

of the triangle OPQ is a fixed straight line.

10. The sides AB and AC of a triangle ABC are given in position

and the harmonic mean between the lengths AB a,nd AC is also given;

prove that the locus of the focus of the parabola touching the sides at

B and C is a circle whose centre lies on the line bisecting the angle

BAG.

11. Parabolas are drawn to touch the axes, which are inclined at

an angle w, and their directrices all pass through a fixed point (h, k).

Prove that all the parabolas touch the straight line

+ ^r-r^ = 1-

h + k sec 0} k + h sec w

CHAPTER XII.

THE ELLIPSE.

247. The ellipse is a conic section in which the

eccentricity e is less than unity.

To find the equation to an ellipse.

Let ZK be the directrix, S the focus, and let SZ be

perpendicular to the directrix.

There will be a point A on SZ, such that

SA = e.AZ (1).

Since e < 1, there will be another point A\ on ZS produced,

such that

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

L. 15

226 COORDINATE GEOMETRY.

Let the length A A! be called 2<a^, and let C be the middle

point of AA', Adding (1) and (2), we have

'la=-AA! = e{AZ^A:Z) = 'l.e.CZ,

i.e. CZ = - (3).

Subtracting (1) from (2), we have

e{A:Z-AZ)^SA'-8A^{SG^CA')-{GA-CS),

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

and hence CS â€” a.e (4).

Let G be the origin, CA' the axis of x, and a line through

C perpendicular to ^^' the axis of 3/.

Let P be any point on the curve, whose coordinates are

X and y, and let Pif be the perpendicular upon the directrix,

and PJV the perpendicular upon A A'.

The focus S is the point (â€” ae, 0).

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

(x + aef-Â¥y^^e^(x+'^\ (Art. 20),

i.e x%l^e') + y' = a\l-e%

a'{l-e')

If in this equation we put x = Oy we have

y = + a\/lâ€”e^j

shewing that the curve meets the axis of y in two points,

B and B', lying on opposite sides of C, such that

B'C ^CB-^a sjT^\ i.e. CB' = CA' - CS\

Let the length CB be called 6, so that

The equation (5) then becomes

Â±-+L = l (6).

a2^b2"-^ ^ ^

i-e. -2 + -2^^i â€” :i^ = l (^)-

THE ELLIPSE. 227

248. The equation (6) of the previous article may be

written

if' Q(? c? â€” x^ {a + x) {a â€” x)

Jf a^ c^ o? '

FN'' AN.NA'

i.e. PJV^ : AN.NA' :: BC : AC\

Def. The points A and A' are called the vertices of

the curve, AA' is called the major axis, and BB' the minor

axis. Also C is called the centre.

249. Since â™¦S' is the point {â€”ae, 0), the equation to

the ellipse referred to S as origin is (Art. 128),

The equation referred to A as origin, and AX and a

perpendicular line as axes, is

(^-^)' , ^' _ 1

a- 0^ a

Similarly, the equation referred to ZX and ZK as axes is,

a

since CZ= â€” ,

(-3'

+ ^- = h

The equation to the ellipse, whose focus and directrix are any

given point and line, and whose eccentricity is known, is easily

written down.

For example, if the focus be the point (-2, 3), the directrix be

the line 2a; + 3?/ + 4 = 0, and the eccentricity be |, the required equa-

tion is

i.e. 261*2 + 181i/2 - 192a;?/ + 1044a; - 2334?/ + 3969 = 0.

15â€”2

228 COORDINATE GEOMETRY.

Generally, the equation to the ellipse, whose focus is the point

(/> P)> whose directrix is Ax + By + C = 0, and whose eccentricity

is e, is

250. There exist a second focus and a second directrix

for the curve.

On the positive side of the origin take a point S', which

is such that SO â€” CS' = ae, and another point Z', such that

e

Draw Z'K' perpendicular to ZZ\ and PM' perpen-

dicular to Z'K'.

The equation (5) of Art. 247 may be written in the

form

x^ â€” 2aex + ah^ + y^ = ^'^^ - 2aeaj + c?,

i.e. {x - aef -^ y^ = e^ (- - x\ ,

i.e. ST^^e^PM"".

Hence any point P of the curve is such that its distance

from S' is e times its distance from Z'K', so that we should

have obtained the same curve, if we had started with S' as

focus, Z'K' as directrix, and the same eccentricity.

251. The sum of the focal distances of any point on the

curve is equal to the major axis.

For (Fig. Art. 247) we have

SP = e.PM, and S'P^e.PM'.

Hence

SP + S'P = e (PM+PM') = e . MM'

= e.ZZ' = 2e.CZ=^2a (Art. 247.)

= the major axis.

Also SP - e . PM^ e.NZ^e.CZ+e.CN-?i + ex',

and S'P - e . PM' = e . NZ' = e . CZ' - e . CN -?i- ex',

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

THE ELLIPSE. LATUS-RECTUM. 229

252. Mechanical construction for an ellipse.

By the preceding article we can get a simple mechanical

method of constructing an ellipse.

Take a piece of thread, whose length is the major axis

of the required ellipse, and fasten its ends at the points S

and aS" which are to be the foci.

Let the point of a pencil move on the paper, the point

being always in contact with the string and keeping the

two portions of the string between it and the fixed ends

always tight. If the end of the pencil be moved about on

the paper, so as to satisfy these conditions, it will trace out

the curve on the paper. For the end of the pencil will be

always in such a position that the sum of its distances from

S and S' will be constant.

In practice, it is easier to fasten two drawing pins at S

and aS", and to have an endless piece of string whose total

length is equal to the sum of SS' and AA'. This string

must be passed round the two pins at S and aS" and then be

kept stretched by the pencil as before. By this second

arrangement it will be found that the portions of the curve

near A and A' can be more easily described than in the first

method.

253. Latus-rectum of the ellipse.

Let LSL' be the double ordinate of the curve which

passes through the focus aS'. By the definition of the curve,

the semi-latus-rectum SL

= e times the distance of L from the directrix

^e.SZ=e{CZ-CS) = e.CZ-e.CS

= a â€” ae^ (by equations (3) and (4) of Art. 247)

= -. (Art. 247.)

254. To trace the curve

%4-' Â«â€¢

230 COORDINATE GEOMETRr.

The equation may be written in either of the forms

Â±Â«yi-s (3).

or X

From (2), it follows that if Q^>a?, i.e. if x> a ov <- a^

then y is impossible. There is therefore no part of the

curve to the right of J.' or to the left of A.

From (3), it follows, similarly, that, if y>h or <:-6,

X is impossible, and hence that there is no part of the curve

above B or below B'.

If X lie between â€” a and + a^ the equation (2) gives two

equal and opposite values for y^ so that the curve is sym-

metrical with respect to the axis of x.

If y lie between â€” h and + 5, the equation (3) gives two

equal and opposite values for x^ so that the curve is sym-

metrical with respect to the axis of y.

If a number of values in succession be given to Â£c, and

the corresponding values of y be determined, we shall

obtain a series of points which will all be found to lie on a

curve of the shape given in the figure of Art. 247.

255. The quantity â€” ^ + -7^ â€” 1 i^ negative^ zero, or

â‚¬(/

positive, according as the point (x , y') lies vjithin, upon, or

without the ellijjse.

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

tlirough Q meet the curve in P, so that, by equation (6) of

Art. 247,

PiP _ 1 ^'

"W ~ ^ â€¢

If Q be within the curve, then y', i.e. QI^, is < FN, so

that

V'' PN^ . -, x'^

0- h- a^

RADIUS VECTOR IN ANY DIRECTION. 231

Hence, in this case,

I.e. â€” r + "v^â€” 1 IS negative.

Similarly, if Q' be without the curve, y' > PN, and then

'2 '2

-^ + ^ - 1 is positive,

a" 6^

256. To find the length of a radius vector from the

centre drawn in a given direction.

The equation (6) of Art. 247 when transferred to polar

coordinates becomes

r^cos^^ r^sin^^

a'W

We thus have the value of the radius vector drawn at any

inclination 6 to the axis.

Since r^ = ^a â€” r-s â€” tit^ â€” r-â€ž-;, , we see that the greatest

62 + (a2-62)sin2(9' *=

value of r is when ^ = 0, and then it is equal to a.

Similarly, B == 90Â° gives the least value of r, viz. h.

Also, for each value of ^, we have two equal and opposite

values of r, so that any line through the centre meets the

curve in two points equidistant from it.

257. Auxiliary circle. Def. The circle which is

described on the major axis, AA\ of an ellipse as diameter,

is called the auxiliary circle of the ellipse.

Let NP be any ordinate of the ellipse, and let it be

produced to meet the auxiliary circle in Q.

Since the angle AQA' is a right angle, being the angle

in a semicircle, we have, by Euc. vi. 8, QN^ = AN. NA'.

232

COORDINATE GEOMETRY.

SO tliat

Hence Art. 248 gives

PN^ : QN' :: BC^ : AG\

PNBCb

QN""AC"a'

Y

^^^^

Q'

y^^^-^^

P

"^^"^^^

f .

P

t^-...

\ C N' N jA' X 1

The point Q in which the ordinate NF meets the

auxiliary circle is called the corresponding point to P.

The ordinates of any point on the ellipse and the

corresponding point on the auxiliary circle are therefore to

one another in the ratio h : Â«, i.e. in the ratio of the

semi-minor to the semi-major axis of the ellipse.

The ellipse might therefore have been defined as follows :

Take a circle and from each point of it draw perpen-

diculars upon a diameter ; the locus of the points dividing

these perpendiculars in a given ratio is an ellipse, of which

the given circle is the auxiliary circle.

258. Eccentric Angle. Def. The eccentric angle

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