Question

Let $P_i$ and $P_i^{\prime }$ be the feet of perpendiculars drawn from foci $S,S^{\prime }$ on a tangent $T_i$ to an ellipse whose length of semi major axis is $20,$ if $\sum _{i=1}^{10}\left(S{P}_i\right)\left(S{P}_i^{\prime }\right)=2560,$ then the value of eccentricity is

• A. $\frac{1}{5}$
• B. $\frac{2}{5}$
• C. $\frac{3}{5}$
• D. $\frac{4}{5}$

Answer 1

The correct option is C $\frac{3}{5}$

$\sum _{i=1}^{10}\left(S{P}_i\right)\left(S{P}_i^{\prime }\right)=2560\Rightarrow 10b^2=2560\Rightarrow b=16$

Given, $a=20$
$\therefore e=\sqrt{1-\frac{b^2}{a^2}}\Rightarrow e=\sqrt{1-\frac{256}{400}}\Rightarrow e=\frac{3}{5}$