Question

The tangent at any point $P$ on a standard ellipse with foci as $S \& S^{\prime}$ meets the tangents at the vertices $A \& A^{\prime}$ in the points $V \& V^{\prime}$, then :

• A. $l( AV ) \cdot l\left( A ^{\prime} V ^{\prime}\right)= b ^2$
• B. $l$ (AV) $l\left( A ^{\prime} V ^{\prime}\right)= a ^2$
• C. $ \angle V ^{\prime} SV =90^{\circ}$
• D. $V^{\prime} S^{\prime} V S$ is a cyclic quadrilateral

Answer 1

Answer: (A), (C), (D)

circle of VV' as diameter is :
$(x-a)(x+a)+\left(y-\frac{b(1-\cos \theta)}{\sin \theta}\right)\left(y-\frac{b(1+\cos \theta)}{\sin \theta}\right)=0$
$\Rightarrow x^2+y^2-2 b \operatorname{cosec} \theta y-\left(a^2-b^2\right)=0$ which passes through $S \& S$