# 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

#### 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