Question

If P is a point on the ellipse of eccentricity $e$ and $A,A^1$ are the vertices and $S,S^1$ are the foci then area of $\Delta SP{S}^1$ : area of $\Delta AP{A}^1$ =

• A. $e^3$
• B. $e^2$
• C. $e$
• D. $\frac{1}{e}$

Answer 1

The correct option is C $e$

Refer to the figure.

Let point P on the ellipse has coordinates $P(x,y)$

$A\left(\Delta SP{S}^{{}^{\prime }}\right)=\frac{1}{2}\times S{S}^{{}^{\prime }}\times y$

$\therefore A\left(\Delta SP{S}^{{}^{\prime }}\right)=\frac{1}{2}\times 2ae\times y$

$\therefore A\left(\Delta SP{S}^{{}^{\prime }}\right)=ae\times y$

$A\left(\Delta AP{A}^{{}^{\prime }}\right)=\frac{1}{2}\times A{A}^{{}^{\prime }}\times y$

$\therefore A\left(\Delta AP{A}^{{}^{\prime }}\right)=\frac{1}{2}\times 2a\times y$

$\therefore A\left(\Delta AP{A}^{{}^{\prime }}\right)=a\times y$

$\frac{A\left(\Delta SP{S}^{{}^{\prime }}\right)}{A\left(\Delta AP{A}^{{}^{\prime }}\right)}=\frac{aey}{ay}$

$\therefore \frac{A\left(\Delta SP{S}^{{}^{\prime }}\right)}{A\left(\Delta AP{A}^{{}^{\prime }}\right)}=e$