Tangent at any point $P$ on an ellipse whose foci are $F_1,F_2$ meets the auxiliary circle of the ellipse at $B_1$, $B_2$. If $F_{1}P+F_{2}P=10$ and $(F_{1}B_{1}) \cdot(F_{2}B_{2})=16$, then eccentricity of the ellipse is equal to?
In this question I understand that $2a=10$ considering the form as $\cfrac{x^2}{a^2}+\cfrac{y^2}{b^2}=1$.What does $(F_{1}B_{1}) \cdot (F_{2}B_{2})=16$ imply?
Please give your suggestions !
$ a = 5, b= 4,$ by the property of perpendicular segments products constancy $ = b^2$
$$ \epsilon = \sqrt{ 1- 16/25} = 3/5 $$
Property constancy of focal ray projections
$$ r_1 r_2 \sin ^2 \psi = b^2 $$
can be easily proved, where $ r_1,r_2 $ are distances to each focus and $\psi$ is tangent-focal ray (mirror reflection) angles at each point.