Q.

Let an ellipse E:x2a2+y2b2=1, a2>b2, passes through 32,1 and has eccentricity 13 If a circle, centered at focus F(α, 0), α > 0, of E and radius 23 intersects E at two points P and Q, then PQ2 is equal to

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a

83

b

163

c

43

d

3

answer is C.

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Detailed Solution

e=13 b2=a2(1-e2)=a21-13 b2=2a23

equation  of ellipse passes through 32,1 is

32a2+1b2=1 32a2+32a2=1 2a2=6  a2=3 b2=2  eq of an ellipse is  x23+y22=1 centre of circle=Focus of ellipse=(1, 0)=(α, 0) radius=23=r

Circle is (x-1)2+y2=43 point of intersection of x23+y22=1 and (x-1)2+y2=43x23+43-(x-1)22=1 2x2+4-3(x-1)2=6 2x2-3x2-3+6x=2 x2-6x+5=0 (x-1) (x-5)=0 x=1 or  5 y2=43(x-1)2 If  x=1  then y2=43y=±23 P1, 23 Q1, -23 PQ=43 PQ2=163

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