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 :

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 ) ⋅ l( A prime V prime)= b 2 (B) l (AV) l( A prime V prime)= a 2 (C)  angle V prime SV =90° (D) V prime S prime V S is a cyclic quadrilateral

Tardigrade Admin · Accepted Answer

circle of VV' as diameter is: (x-a)(x+a)+(y-(b(1- cos θ)/ sin θ))(y-(b(1+ cos θ)/ sin θ))=0 ⇒ x2+y2-2 b operatornamecosec θ y-(a2-b2)=0 which passes through S S

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

$l (A V) \cdot l (A^{'} V^{'}) = b^{2}$

$l$ (AV) $l (A^{'} V^{'}) = a^{2}$

$∠ V^{'} S V = 9 0^{\circ}$

$V^{'} S^{'} V S$ is a cyclic quadrilateral

circle of VV' as diameter is :
$(x - a) (x + a) + (y - \frac{b ( 1 - c o s θ )}{s i n θ}) (y - \frac{b ( 1 + c o s θ )}{s i n θ}) = 0$
$\Rightarrow x^{2} + y^{2} - 2 b cosec θ y - (a^{2} - b^{2}) = 0$ which passes through $S & S$

Q. The tangent at any point P on a standard ellipse with foci as S&S′ meets the tangents at the vertices A&A′ in the points V&V′, then :

l(AV)⋅l(A′V′)=b2

l (AV) l(A′V′)=a2

∠V′SV=90∘

V′S′VS is a cyclic quadrilateral