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Question

A circle and an ellipse have centres at (0,0) and the circle passes through foci F1 and F2 of the ellipse, such that the two curves intersect at four points. Let P be any one of their points of intersection. If the length of major axis of the ellipse is 17 and area of the triangle PF1F2 is 30, then distance between foci is

A
11
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B
12
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C
13
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D
15
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Solution

The correct option is C 13

Assume horizontal ellipse.
Radius of circle =ae
C:x2+y2=a2e2
E:x2a2+y2b2=1

So, Area(PF1F2)=12∣ ∣ae01ae01xy1∣ ∣=±30

Expand about 2nd column
y(ae+ae)=±60
y=30ae

C:x2=a2e2y2
E:x2=a2(1y2b2)

a2(1y2b2)=a2e2y2

a2(1900a2e2b2)=a2e2900a2e2

a2(1e2)=900e2b2900a2e2

a2(1e2)=900e2[1a2(1e2)1a2]

(a2(1e2))2=900
1e2=30a2

e2=130289×4

e=1317
Distance between focii is 2ae=13

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