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S. L. (Sidney Luxton) Loney.

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of any point F on the ellipse is the angle NCQ made with
the major axis by the straight line CQ joining the centre G
to the point Q on the auxiliary circle which corresponds to
the point P.

This angle is generally called ^.



THE ECCENTRIC ANGLE. 233

We have CJV=CQ . cos, <f> = a cos (f>,

and J^Q = CQ sin (f> = a sin <^.

Hence, by the last article,

a

The coordinates of any point P on the ellipse are there-
fore a cos <f> and b sin ^.

Since F is known when ^ is given, it is often called
" the point <^."

259. To obtain the equation of the straight line joi^iing
two points on the ellij^se whose eccentric angles are given.

Let the eccentric angles of the two points, P and P\ be
ff> and ^', so that the points have as coordinates

{a cos ^, b sin ^) and (a cos <^', 6 sin ^f)').

The equation of the straight line joining them is

, . , 6 sin d)' - 6 sin </> ,

2/ - 6 sm <h -, V (x — a cos d))

a cos fp —a cos ^ ^ '

_b 2cosl(<^ + <^>inl(</> - c^)
a 'Ssin !(<}!> + </,') sin i(<^-</>y'^ «cos<^;

6 cosl(<^ + <^') . .

= - - . -i— f-h , — 7\ {x-a cos </>),



a ' sin J (^' + <^)



i.e.

a; (ft + cb' y . ch + ch' 4> + ^' . .6 + 6'

- cos — ~ — + Y- Sin -^~ - = cos <i> cos ^ + sin </> sm ^
C&262 2 ^2

r «^ + </,'-l <^-c^' ,-,

-cos|^<^ - ^^-^J=:cos^^l-^ (1).

This is the required equation.

Cor. The points on the auxiliary circle, corresponding to P and
P', have as coordinates (a cos 0, a sin 0) and (a cos 0', a sin 0')-

The equation to the line joining them is therefore (Art. 178)

X <t> + <i) V . (i> + (h' (i>-<t>'

- cos ^— -^ + •- sin - ^ =cos —r- •
a 2 a 2 2



234 COORDINATE GEOMETRY.

This straight line and (1) clearly make the same intercept on the
major axis.

Hence the straight line joining any two points on an ellipse, and
the straight line joining the corresponding points on the auxiliary
circle, meet the major axis in the same point.



EXAMPLES. XXXII.

1. Find the equation to the ellipses, whose centres are the
origin, whose axes are the axes of coordinates, and which pass
through (a) the points (2, 2), and (3, 1),

and (/3) the points (1, 4) and (-6, 1).

Find the equation of the ellipse referred to its centre

2. whose latus rectum is 5 and whose eccentricity is |,

3. whose minor axis is equal to the distance between the foci and
whose latus rectum is 10,

4. whose foci are the points (4, 0) and ( - 4, 0) and whose
eccentricity is ^.

5. Find the latus rectum, the eccentricity, and the coordinates
of the foci, of the ellipses

(1) a:^ + Sy^ = a^, (2) 5x'^ + 4y'^ = l, and (3) 9x^ + 5y^-S0y = O.

6. Find the eccentricity of an ellipse, if its latus rectum be equal
to one half its minor axis.

7. Find the equation to the ellipse, whose focus is the point
(-1, 1), whose directrix is the straight line x -y + S=0, and whose
eccentricity is ^.

8. Is the point (4, - 3) within or without the ellipse

5x^ + 7y^ = lV?

9. Find the lengths of, and the equations to, the focal radii drawn
to the point (4 ^3, 5) of the ellipse

25x2 + 162/2=1600.

10. Prove that the sum of the squares of the reciprocals of two
perpendicular diameters of an ellipse is constant.

11. Find the inclination to the major axis of the diameter of the
ellipse the square of whose length is (1) the arithmetical mean,
(2) the geometrical mean, and (3) the harmonical mean, between the
squares on the major and minor axes.

12. Find the locus of the middle points of chords of an ellipse
which are drawn through the positive end of the minor axis.

13. Prove that the locus of the intersection of AP with the
straight line through A' perpendicular to A'F is a straight line which
is perpendicular to the major axis.



[EXS. XXXII.] THE ECCENTRIC ANGLE. 235

14. Q is the point on the auxiliary circle corresponding to P on
the ellipse; PLM is drawn parallel to CQ to meet the axes inL andilf ;
prove that PL = b and PM—a.

15. Prove that the area of the triangle formed by three points on
an ellipse, whose eccentric angles are ^, 0, and \{/, is

. , . 0-^ . yl/-d . d-(t>
lab sin -^-^ sm i-^— sm — ^ .

Prove also that its area is to the area of the triangle formed by the
corresponding points on the auxiliary circle as 6 : a, and hence that
its area is a maximum when the latter triangle is equilateral, i.e. when

27r
0-^ = ^-0 = —.

16. Any point P of an ellipse is joined to the extremities of the
major axis; prove that the portion of a directrix intercepted by them
subtends a right angle at the corresponding focus.

17. Shew that the perpendiculars from the centre upon all chords,
which join the ends of perpendicular diameters, are of constant
length.

18. If a, j8, 7, and 5 be the eccentric angles of the four points of
intersection of the ellipse and any circle, prove that

a + /3 + 7 + 5isan odd multiple
of IT radians.

[See Trigonometry, Part II, Art. 31, and Page 37, Ex. 15.]

19. The tangent at any point P of a circle meets the tangent at a
fixed point A in T, and T is joined to B, the other end of the
diameter through A ; prove that the locus of the intersection of AP

and BT is an ellipse whose eccentricity is — .- .

20. From any point P on the ellipse, PN is drawn perpendicular
to the axis and produced to Q, so that NQ equals PS, where ^ is a
focus ; prove that the locus of Q is the two straight lines y±ex + a = 0.

21. Given the base of a triangle and the sum of its sides, prove
that the locus of the centre of its incircle is an ellipse.

22. With a given point and line as focus and directrix, a series
of ellipses are described; prove that the locus of the extremities of
their minor axes is a parabola.

23. A line of fixed length a + b moves so that its ends are always
on two fixed perpendicular straight lines; prove that the locus of a
point, which divides this line into portions of length a and b, is an
ellipse.

24. Prove that the extremities of the latera recta of all ellipses,
having a given major axis 2a, lie on the parabola x^= -a{y- a).



236 COORDINATE GEOMETRY,

260. Tojind the intersections of any straight li7ie with
the ellipse

T + |I=1 (1).

Let the equation of the straight line be

y ^ TThx + c (2).

The coordinates of the points of intersection of (1) and
(2) satisfy both equations and are therefore obtained by
solving them as simultaneous equations.

Substituting for y in (1) from (2), the abscissae of the
points of intersection are given by the equation

^ (mx + cf

a' '^ ^ir~ " '

i.e. x" {a^mP + b') + 2a^mcx + a' (c" - If) = (3).

This is a quadratic equation and hence has two roots,
realj coincident, or imaginary.

Also corresponding to each value of x we have from (2)
one value of y.

The straight line therefore meets the curve in two points
real, coincident, or imaginary.

The roots of the equation (3) are real, coincident, or
imaginary according as

(2a^?7ic)^— 4 (b^+a^ni^) x a^ (d^—lr) is positive, zero, or negative,

i.e. according as h^(h'^-\-a^m?)—l?c^ is positive, zero, or negative,

i.e. according as c^ is < =: or > ahn^ + h^.

261. To find the length of the chord intercepted hy the
ellipse on the straight line y = mx + c.

As in Art. 204, we have

Marine , or {c^ — Ir)

X-, -f- x^ ^= — ~ 5 Yh J ^nci X1X2 =^ s 5 7 „ ,



, , , 2a6 sja-nr + b' — c

so that X, — Xo = 7,-7, — Th

a^mr + o"



EQUATION TO THE TANGENT 237

The length of the required chord therefore

= J{x^ - flJa)^ + (2/1 - 2/2)^ = {^1 -x^ sll+m^

2ab \/l + m^ Ja^m^ + b^ — c^
a^m^ + b^

262. To find the equation to the tangent at any 'point
{x\ y') of the ellipse.

Let P and Q be two points on the ellipse, whose coordi-
nates are {x\ y') and {x", y").

The equation to the straight line PQ is

y-2/'=J^|'(^-*') (!)•

Since both P and Q lie on the ellipse, we have

10 ro



a-'¥=' • (2)'



,2



-^^+v=l (^>-



Hence, by subtraction,

x"'-x'^ y"^-y''_^
a^ "^ 6^ ~ '

( y"-y){y" + y') _ _ { x"-x'){x" + x')

y"-y' _b^ x" + x'
**^' ^^'^^''" a''y" + y''

On substituting in (1) the equation to any secant PQ
becomes

f 0'' X -T X . ,v J .^

y-y- 5-77 ,{x~x) (4).

To obtain the equation to the tangent we take Q
indefinitely close to P, and hence, in the limit, we put
a;" = X and y" = y ,



238 COORDINATE GEOMETRY.

The equation (4) then becomes

/ ox, •>



a" 0" c? h^



i.e. -2 + -.T = -2 + y^ = 1' % equation (2).



The required equation is therefore

a2 "*■ b2 " ■*■•

Cor. The equation to the tangent is therefore ob-
tained from the equation to the curve by the rule of
Art. 152.

263. To find the equation to a tangent in terms of the
tcmgent of its inclination to the major axis.

As in Art. 260, the straight line

y = 7nx + c (1)

meets the ellipse in points whose abscissae are given by

af (62 + ahn") + Imca'x + a" (c^ - h'') = 0,

and, by the same article, the roots of this equation are
coincident if

c = \ja?m^ + y^.

In this case the straight line (1) is a tangent, and
it becomes

y = mx+ Va2m2 + b2 (2).

This is the required equation.

Since the radical sign on the right-hand of (2) may
have either + or — prefixed to it, we see that there are two
tangents to the ellipse having the same w^, i.e. there are
two tangents parallel to any given direction.

The above form of the equation to the tangent may be deduced
from the equation of Art. 262, as in the ease of the parabola
(Art. 206). It will be found that the point of contact is the point

/ - c?'m y^ \



EQUATION TO THE TANGENT. 239

264. By a proof similar to that of the last article, it
may be shewn that the straight line

X cos a + y sin a =^p

touches the ellipse, if

p2 = a2 cos2 a + b2 sin2 a.

Similarly, it may be shewn that the straight line
Ix + my = n
touches the ellipse, if aH^ + b^m^ = v?.

265. Equation to the tangent at the point whose
eccentric angle is <f>.

The coordinates of the point are (a cos ^, b sin (ft).

Substituting x' = a cos <^ and y' — b sin <f> in the equation
of Art. 262, we have, as the required equation,

X V

- COS ^ + ^ sin ^ = 1 (1).

3> D

This equation may also be deduced from Art. 259.

For the equation of the tangent at the point "<^" is
obtained by making cf>' = (ft in the result of that article.

Ex. Find the intersection of the tangents at the points <p and <p'.
The equations to the tangents are

- COS + V sin - 1 = 0,
a

and - cos 0' + ^ sin 0' - 1 = 0.

a b

The required point is found by solving these equations.
We obtain

X y '

a b -1 1



sin - sin 0' cos <p' - cos <p sin 0' cos - cos <p' sin sin (0 - 0') '

i.e.

5 y 1



(b + <b . (b — d)' ^, . + 0'. (b — <b ^ , (h — d) dt — <b

2a COS ^ sin ^ ^ 2b sin ^ ■ sm - 2 sin ~^ cos ■ ^



240 COORDINATE GEOMETRY.

Hence ^=a '=-2iiM±£) and j, = i,!i^i <|±J3.

COS i{(f>-^') COS J (0 - <p')

266. Equation to the normal at the point {x, y').

The required normal is the straight line which passes
through the point {x\ y) and is perpendicular to the
tangent, i.e. to the straight line

a'y y

Its equation is therefore

y — y' —m(x — x')^

where m ( — ^-\ = -l, i.e. m ^ ^, , (Art. 69).

\ ay / o-'x

ft u

The equation to the normal is therefore y — y' = -— -, {x - x),

X - X' y - y



^.e.



a2 b2



267. Equation to the normal at the point whose eccentric
angle is <j>.

The coordinates of the point are a cos <^ and b sin (f>.

Hence, in the result of the last article putting

X —a cos ^ and y =h sin <j!>,

. , , x — a cos c& y — h sin <i>

it becomes , =: . , ,

cos <p sm <^

a h

ax „ by
cos <p sm ip

The required normal is therefore

ax sec ^ — by cosec ^ = a2 — b2.



SUBTANGENT AND SUBNORMAL.

# 268. Equation to the normal in the form y = mx + c.
The equation to the normal at {x', y') is, as in Art. 266,



241






_ , a^u ^- ^ X ay

Let^, = m,sothat- = ^-^^.



Hence, since {x', y') satisfies the relation -^ + ^ = 1> ^^^ obtain



y"^



62



b^m



y ~ Ja^ + hHi^'
The equation to the normal is therefore

y=mx- f- ^.

This is not as important an equation as the corresponding equa-
tion in the case of the parabola. (Art. 208.)

When it is desired to have the equation to the normal expressed
in terms of one independent parameter it is generally better to use
the equation of the previous article.

269. To find the length of the suhtangent and sub-
normal.




Let the tangent and normal at P, the point (x', 3/'),
meet the axis in T and G respectively, and let PN be the
ordinate of P,



L.



16



242 COORDINATE GEOMETRY.

The equation to the tangent at P is (Art. 262)

? + f^=l (!)•

To find where the straight line meets the axis we put
y — and have

x=-, i.e. CT^



^ ' CN'-



I.e.



GT.GN=a^=GA^ (2).



Hence the subtangent NT



a? , a^ — x'^



^CT-GN=^-x'^

X X

The equation to the normal is (Art. 266)

X — X y ~y
x y' '

To find where it meets the axis, we put 2/ — 0, and have

X X y 7 ■>

^^^^~ / ~ ^ J

x_ ^

'a? ¥

i.e. Ga=^x = x'-^,x'='^^x'=^eKx'^e\GN...{3).

CO CO

Hence the subnormal JVG

^GN-GG^{l-e^)GN,

i.e. NG::NGv.l~e^'.l

:: h" : a\ (Art. 247.)

Cor. If the tangent meet the minor axis in t and Pn
be perpendicular to it, we may, similarly, prove that

Gt.Gn^h\

270. Some properties of the ellipse.

(a) SG = e,SP, and the tangent and normal at P bisect the
external and internal angles between the focal distances of P.

By Art. 269, we have CG=e^x'.



SOME PROPERTIES OF THE ELLIPSE. 243

Hence SG = SC+CG = ae + e^x' = e.SP, by Art. 251.

Also S'G=zCS'-CG = e{a-ex') = e.S'P.

Hence SG : S'G :: SP : S'P.

Therefore, by Euc. vi, 3, PG bisects the angle SPS'.

It follows that the tangent bisects the exterior angle between
SP and ST.

(/3) If SY and S'Y' be the perpendiculars from the foci upon the
tangent at any point P of the ellipse, then Y and Y' lie on the auxiliary
circle, and SY . S'Y' = b'^. Also GY and S'P are parallel.

The equation to any tangent is

X co^a + y Bva.a=p (1),



where p = sja^ cos^ a + 6^ gin^ a (Art. 264).

The perpendicular SY to (1) passes through the point {-ae, 0)
and its equation, by Art. 70, is therefore

(a; + ae)sin a- j/cos a = (2).

If Y be the point {h, k) then, since Y lies on both (1) and (2), we
have

h cos a + k sin a — sja^ cos^ a + b^ sin^ a,

and h sin a - ^ cos a= -ae sin a = - ija^ - b^ sin^ a.

Squaring and adding these equations, we have h'^ + k'^ = a'^, so that
Y lies on the auxihary circle x^ + y'^:= c^.

Similarly it may be proved that Y' lies on this circle.
Again S is the point ( - ae, 0) and S' is (ae, 0).
Hence, from (1),

SY=p-\-ae cos a, and S'Y'=p - ae cos a. (Art. 75.)
Thus SY . S'Y' =^2 _ a2g2 cos2 a

= a^ cos^ a + &2 sin^ a-{a'^- b^) cos^ a



Also CT= "'



GN'
a^ _a{a-eCN)



and therefore S'T=~-^-ae= ^,,

ON CN

GT «^__CT

''■ S'T~ a-e.G~N~ S'P'

Hence GY and S'P are parallel. Similarly GY' and SF are
parallel.

16—2



244 COORDINATE GEOMETRY.

(7) If the normal at any point P meet the major and minor axes
in G and g, and if GF he the perpendicular upon this normal, then
FF.PG^ &2 aj^^ PF.Pg = a2.

The tangent at any point P (the point " 0") is
ah



Hence Pi^= perpendicular from G upon this tangent
1 ah



V



cos^ sin^ (p fjb^ cos^ + a'-^ sin^



(1).



The normal at P is

-^ - X=a^-h- (2).

cos sm

If we put y = 0, we have GG = cos 0.

a cos - cos J + 62 sin3

h^

= "2 COS^ + &2 sin^ 0,



i.e. PG^-Jb^cos^(p + a^sin^(p.

a ^

From this and (1), we have PF.PG = h^.

If we put a; = in (2), we see that g is the point

(0, ^sin0j.

Hence Pg^ = a^cos^(p+i h sin 04 ^ — sin j ,

so that ^^~h '^^^ ^^^^ + ^2 sin^ 0.

From this result and (1) we therefore have

PF. Pg = a\

271. To find the locus of the point of intersectio7i of
tangents which meet at right angles.

Any tangent to the ellipse is

y — 7nx + sja^rri^ + 6'^,
and a perpendicular tangent is



y - in''^\J'''{-^3*^^-



TANGENTS AND NORMALS. EXAMPLES. 245

Hence, if {h, k) be their point of intersection, we have

k — mil — sjahii? + fe- (1),

and ink + A = sjo? + Ir'nn? (2).

If between (1) and (2) we eliminate m, we shall have a
relation between h and k. Squaring and adding these
equations, we have

(P + W) (1 + m^) = (^2 + W) (1 + m%

i.e. h'' + k^ = a^ + h\

Hence the locus of the point (Ji, k) is the circle

a^ + 2/^ = «- + h^,

i.e. a circle, whose centre is the centre of the ellipse, and
whose radius is the length of the line joining the ends
of the major and minor axis. This circle is called the
Director Circle.



EXAMPLES. XXXIII.

Find the equation to the tangent and normal

1. at the point (1, f) of the ellipse 4a;2 + Qt/^ = 20,

2. at the point of the ellipse 5a;2 + 3z/2 = 137 whose ordinate is 2,

3. at the ends of the latera recta of the ellipse ^x^ + 16?/2 = 144.

4. Prove that the straight line y = x + ^^^ touches the ellipse

5. Find the equations to the tangents to the ellipse 4j,'- + 3?/^ = 5
which are parallel to the straight line y = Sx + 7.

Find also the coordinates of the points of contact of the tangents
which are inclined at 60° to the axis of x.

6. Find the equations to the tangents at the ends of the latera
recta of the ellipse -g + T2 =^ •'■» ^^^ shew that they pass through the
intersections of the axis and the directrices.

7. Find the points on the ellipse such that the tangents there
are equally inclined to the axes. Prove also that the length of the
perpendicular from the centre on either of these tangents is

2 '



x/'



246 COORDINATE GEOMETRY. [ExS.

8. In an ellipse, referred to its centre, the length of the sub-
tangent corresponding to the point (3, V) is -V-; prove that the
eccentricity is f .

9. Prove that the sum of the squares of the perpendiculars on
any tangent from two points on the minor axis, each distant jja?' - fe^
from the centre, is ^a?-.

10. Find the equations to the normals at the ends of the latera
recta, and prove that each passes through an end of the minor axis if

e4 + e2 = l.

11. If any ordinate MP meet the tangent at .L in Q, prove that
MQ and SP are equal.

12. Two tangents to the ellipse intersect at right angles; prove
that the sum of the squares of the chords which the auxiliary circle
intercepts on them is constant, and equal to the square on the line
joining the foci.

13. If P be a point on the ellipse, whose ordinate is y', prove
that the angle between the tangent at P and the focal distance of P

is tan~i — ; .
aey

14. Shew that the angle between the tangents to the ellipse

— \.^ = '\. and the circle x~ + y'^ = ab at their points of intersection is
a^ Ir

a— b

tan~^ ~7=? .
y/ab

15. A circle, of radius r, is concentric with the ellipse ; prove
that the common tangent is inclined to the major axis at an angle

tan~i X I —, H and find its length,

16. Prove that the common tangent of the ellipses

a;2 y'^ _'ix T x^ y^ 2x _
a^ b^ c />- a'^ c

subtends a right angle at the origin.

17. Prove that PG.Pg = SP. ST, and CG.CT= CS^.

18. The tangent at P meets the axes in T and t, and CY is the
perpendicular on it from the centre; prove that (1) Tt . PY=a - b'^,
and (2) the least value oiTtisa + b.

19. Prove that the perpendicular from the focus upon any tangent
and the line joining the centre to the point of contact meet on the
corresponding directrix.

20. Prove that the straight lines, joining each focus to the foot of
the perpendicular from the other focus upon the tangent at any
point P, meet on the normal at P and bisect it.



XXXIIIJ TANGENTS AND NOKMALS. EXAMPLES. 247

21. Prove that the circle on any focal distance as diameter touches
the auxiliary circle.

22. Find the tangent of the angle between CP and the normal at

P, and prove that its greatest value is ^ , .

2a6

23. Prove that the straight line lx + my=n is a normal to the

a^ b'~ (a^ - 6^)2
fillipse, if i2 + — 2 = 9- •

24. Find the locus of the point of intersection of the two straight

lines - - l + t = and - + ^ - 1 = 0.
a a

Prove also that they meet at the point whose eccentric angle is
2tan-ii.

25. Prove that the locus of the middle points of the portions of
tangents included between the axes is the curve

26. Any ordinate NP of an ellipse meets the auxiliary circle in
Q ; prove that the locus of the intersection of the normals at P and
Q is the circle x^ + y^ = {a + h) ^.

27. The normal at P meets the axes in G and g ; shew that the
loci of the middle points of PG and Gg are respectively the ellipses

4^.2 4^,2

^2(1^,2)2 + |- = 1, anda%2 + 62^2^|(,,2_62)2.

28. Prove that the locus of the feet of the perpendicular drawn
from the centre upon any tangent to the ellipse is

r2 = a2 cos2 ^ + &2 sin2 e. [ Use Art. 264.]

29. If a number of ellipses be described, having the same major
axis, but a variable minor axis, prove that the tangents at the ends of
their latera recta pass through one or other of two fixed points.

30. The normal GP is produced to Q, so that GQ = n. GP.

Prove that the locus of Q is the ellipse -rn » jtck + -^o = 1.

a^{n + e^-ne^y n%^

31. If the straight line y = mx + c meet the ellipse, prove that the
equation to the circle, described on the line joining the points of
intersection as diameter, is

(a2m2 + 62) (a;2 + y2) + 2ma^cx - 2b'^cij + c^ {a^ + 62) - a262 (1 + wF) = 0.

32. PM and PN are perpendiculars upon the axes from any point
P on the ellipse. Prove that MN is always normal to a fixed
concentric ellipse.



248 COORDINATE GEOMETRY. [EXS. XXXIII.]

33. Prove that the sum of the eccentric angles of the extremities
of a chord, which is drawn in a given direction, is constant, and
equal to twice the eccentric angle of the point at which the tangent is
parallel to the given direction.

34. A tangent to the ellipse ~2 + fa — -^ meets the ellipse

in the points P and Q; prove that the tangents at P and Q are at
right angles.

272. To prove that through any given point (cci, 3/1)
there pass, in general, two tangents to an ellipse.

The equation to any tangent is (by Art. 263)

y = mx + sja^m^ + b^ (1).

If this pass through the fixed point (x-^^, y-^, we have
2/1 — mcci = sjo^m? + IP-,
i.e. y^ — Imx-^y^ + w^x^ — c^w? + IP",
i.e. m^{x^-a^)-'imXjy^+{y{ - ¥)^Q (2).

For any given values of x^ and y^ this equation is in
general a quadratic equation and gives two values of m
(real or imaginary).

Corresponding to each value of m we have, by sub-
stituting in (1), a different tangent.

The roots of (2) are real and different, if

(- 2x^y^y - 4 (a?!^ — a^) {y^ - If) be positive,

i. e. if IPx^ + a^y^ — a^lP be positive,

X " 11
i.e. ii ^ + ^ - 1 be positive,

a^ 0^

i.e. if the point {x^, y-^ be outside the curve.

The roots are equal, if

h-x^^ + a'^y^^ - a-62

be zero, i.e. if the point {x^, y-^ lie on the curve.



CHORD OF CONTACT OF TANGENTS. 249

The roots are imaginary, if

be negative, i.e. if the point (a?i, y^ lie within the curve
(Art. 255).

273. Equation to the chord of contact of tangents
drawn from a point {x^, y-^).

The equation to the tangent at any point Q, whose
coordinates are x and y', is

^ + ^' =, 1

Also the tangent at the point R, whose coordinates are
x" and y\ is

H It

a^ Jy"

If these tangents meet at the point 57, whose coordi-
nates are x^ and y^, we have

a^ 62 -^ l^^

and ^' + 2/_|:^l ^2).

a^ 0^ ^ ^

The equation to ^^ is then

^^^=^ (3)-

For, since (1) is true, the point {x, y') lies on (3).

Also, since (2) is true, the point {x'\ y") lies on (3).

Hence (3) must be the equation to the straight line
joining (x', y) and {x\ y"), i.e. it must be the equation to
QR the required chord of contact of tangents from {x^, v/i)-

274. To find the equation of the polar of the point
(•''^1) 2/i) '^'^'i^t' Tespect to the ellip)se

S-J=l- [Art. 162.]



250 COORDINATE GEOMETRY.

Let Q and M be the points in which any chord drawn
through the point (ccj, y-^ meets the ellipse [Fig. Art. 214].

Let the tangents at Q and R meet in the point whose
coordinates are (A, h).

"We require the locus of {Ji, k).

Since QR is the chord of contact of tangents from
(A, ^), its equation (Art. 273) is

xh yk
~^'^'¥^

Since this straight line passes through the point (x^, ^j),
we have

hx^ ky^



a



+ ^ = 1 (!)•



Since the relation (1) is true, it follows that the point
(A, k) lies on the straight line

^^W=^ (^)-

Hence (2) is the equation to the polar of the point

Cor. The polar of the focus (ae, o) is


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