Question

An ellipse has foci $\left(4,2\right),\left(2,2\right)$ and it passes through the point $P\left(2,4\right)$ . The eccentricity of the ellipse is

• A. $tan\frac{Ï€}{10}$
• B. $tan\frac{Ï€}{12}$
• C. $tan\frac{Ï€}{6}$
• D. $tan\frac{Ï€}{8}$

Answer 1

Answer: (D)

Let, $\left(4,2\right)=S_1$ and $\left(2,2\right)=S_2$ and eccentricity of ellipse is $e$
then $S_1{S}_2=2ae$ and $P{S}_1+P{S}_2=2a$
(where $2a$ is length of major axis)
$⇒e=\frac{S_1{S}_2}{P{S}_1+P{S}_2}=\frac{2}{2\sqrt{2}+2}$
$⇒e=\frac{1}{\sqrt{2}+1}=\sqrt{2}−1=tan\frac{π}{8}$