A circle has the same center as an ellipse and passes through the foci `F_1a n dF_2` of the ellipse, such that the two cuves intersect at four points. Let `P` be any one of their point of intersection. If the major axis of the ellipse is 17 and the area of triangle `P F_1F_2` is 30, then the distance between the foci is 13 (b) 10 (c) 11 (d) none of these

Options

A. 13
B. 10
C. 11
D. non of these

Explanation

Let the ellipe be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` and the circle be `x^(2)+y^(2)=a^(2)e^(2)`
Radius of circle =ae
One of the points of intersection of the circle and the ellipse is `((a)/(e)sqrt(2e^(2)-1),(a)/(e)(1-e^(2)))`
Now, area of `DeltaPF_(1)F_(2)=(1)/(2)|{:((a)/(e)sqrt(e^(2)-1),(a)/(e)(1-e^(2)),1),(ae,0,1),(-ae,0,1):}|=30`
`or(1)/(2)|(a)/(e)(1-e^(2))(2ae)|=30`
or `a^(2)(1-e^(2))=30`
or `a^(2)e^(2)=a^(2)-30=((17)/(2))^(2)-30=(169)/(4)`
or 2ae=13