Question

Let $A(acos\theta ,bsin\theta )$ is a variable point, $S=(ap,0)$ and $S^{\prime }\equiv (-ap,0)$ are two fixed points where $p=\sqrt{\frac{a^2-b^2}{a^2}}$. If the locus of Incentre of triangle ASS' is a conic then the eccentricity of the conic in terms of p is

• A. $\sqrt{\frac{2p}{1+p}}$
• B. $\sqrt{\frac{p}{1+p}}$
• C. $\sqrt{\frac{1-p}{1+p}}$
• D. $\frac{p}{2(1+p)}$

Answer 1

The correct option is A $\sqrt{\frac{2p}{1+p}}$

A is any point on the ellipse
S, S' are foci SS' = 2ae
SA $=a(1-ecos\theta )$
S’A $=a(1+ecos\theta )$
In centre (x, y)
$x=apcos\theta ,y=\frac{bp}{1+p}sin\theta$