Question

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

Answer 1

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