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The Eccentricity of the Ellipse, If the Distance Between the Foci is Equal to the Length of the Latus-rectum, is - Mathematics

MCQ
Sum

The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is

Options

  • \[\frac{\sqrt{5} - 1}{2}\]

     

  • \[\frac{\sqrt{5} + 1}{2}\]

     

  • \[\frac{\sqrt{5} - 1}{4}\]

     

  • none of these

     
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Solution

\[e = \frac{\sqrt{5} - 1}{2} \]
According to the question, the distance between the foci is equal to the length of the latus rectum.
\[\frac{2 b^2}{a} = 2ae\]
\[ \Rightarrow b^2 = a^2 e\]
\[\text{ Now, }e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{a^2 e}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - e}\]
On squaring both sides, we get:
\[ e^2 + e - 1 = 0\]
\[ \Rightarrow e = \frac{- 1 \pm \sqrt{1 + 4}}{2}\]
\[ \Rightarrow e = \frac{\sqrt{5} - 1}{2} \left( \because\text{ e cannot be negative }\right)\]

Concept: Introduction of Ellipse
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 26 Ellipse
Q 5 | Page 28
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