Question

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

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

Answer 1

Answer: (C)

$\frac{3}{5}$

Given: ${\sum }_{i=1}^{10}\left(S{P}_i\right)\left(S{{}^{\prime }P}_i^{\prime }\right)=2560$

we know that $SP.S^{\prime }{P}^{\prime }$ to any tangent $T$ of the ellipse$=b^2$

$\Rightarrow 10b^2=2560$

$\Rightarrow b^2=256$

Given length of semi major axis is $20$

so, $e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{\frac{400-256}{400}}=\frac{12}{20}=\frac{3}{5}$