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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Ellipse

NTA Abhyas 2020 · Mathematics · Ellipse

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

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

Options

  1. Atanπ10tan\dfrac{\pi}{10}
  2. Btanπ12tan\dfrac{\pi}{12}
  3. Ctanπ6tan\dfrac{\pi}{6}
  4. Dtanπ8tan\dfrac{\pi}{8}

Correct answer

D. tanπ8tan\dfrac{\pi}{8}

Step-by-step solution

Let,& (4,2)=S1and(2,2)=S2\left(4, 2\right) = {S}_{1} and \left(2, 2\right) = {S}_{2} and eccentricity of ellipse is ee
then S1S2=2aeandPS1+PS2=2a{S}_{1}{S}_{2} = 2ae and {PS}_{1} + {PS}_{2} = 2a
(where 2a2a is~length of major axis)
⇒e=S1S2PS1+PS2=222+2⇒ e = \dfrac{{S}_{1}{S}_{2}}{P{S}_{1} + P{S}_{2}} = \dfrac{2}{2\sqrt{2} + 2}
⇒e=12+1=2−1=tanπ8⇒ e = \dfrac{1}{\sqrt{2} + 1} = \sqrt{2} - 1 = tan\dfrac{\pi}{8}
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