Question

The tangent at $\alpha$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at two points which subtend a right angle at the centre. Then e =

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Equation of tangent at $\alpha$ is $\frac{x}{a}\cos \alpha +\frac{y}{b}\sin \alpha =1$
Equation of auxillay circle is $x^2+y^2=a^2$
Combined equation is $x^2+y^2=a^2{(\frac{x}{a}\cos \alpha +\frac{y}{b}\sin \alpha )}^2$
Since these lines are perpendicular then $1-{\cos }^2\alpha +1-\frac{a^2}{b^2}{\sin }^2\alpha =0\Rightarrow 1+{\sin }^2\alpha ={\sin }^2\alpha \left(\frac{a^2}{b^2}\right)$
$\Rightarrow \frac{b^2}{a^2}=\frac{{\sin }^2\alpha }{1+{\sin }^2\alpha }\Rightarrow 1-e^2=\frac{{\sin }^2\alpha }{1+{\sin }^2\alpha }\Rightarrow e^2=\frac{1}{1+{\sin }^2\alpha }\Rightarrow e=\frac{1}{\sqrt{1+{\sin }^2\alpha }}$