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JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Ellipse

JEE Main · Mathematics · Ellipse

An ellipse has foci (4,2), (2,2) and it passes through the point P...

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

Options

A $tan\dfrac{\pi}{10}$
B $tan\dfrac{\pi}{12}$
C $tan\dfrac{\pi}{6}$
D $tan\dfrac{\pi}{8}$

Correct answer

D. $tan\dfrac{\pi}{8}$

Step-by-step solution

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

{PS}_{1} + {PS}_{2} = 2a

(where

2a

is length of major axis)

⇒ e = \dfrac{{S}_{1}{S}_{2}}{P{S}_{1} + P{S}_{2}} = \dfrac{2}{2\sqrt{2} + 2}

⇒ e = \dfrac{1}{\sqrt{2} + 1} = \sqrt{2} - 1 = tan\dfrac{\pi}{8}

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