# Find the Equation of the Ellipse in the Standard Form Whose Minor Axis is Equal to the Distance Between Foci and Whose Latus-rectum is 10. - Mathematics

Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus-rectum is 10.

#### Solution

$\text{ According to the question, the minor axis is equal to the distance between the foci }.$
$\text{ i . e } . 2b = 2\text{ ae and } \frac{{2b}^2}{a} = 10 \text{ or } b^2 = 5a$
$\Rightarrow b = ae$
$\Rightarrow b^2 = a^2 e^2$
$\Rightarrow b^2 = a^2 \left( 1 - \frac{b^2}{a^2} \right) \left( \because e = \sqrt{1 - \frac{b^2}{a^2}} \right)$
$\Rightarrow b^2 = a^2 - b^2$
$\Rightarrow a^2 = 2 b^2$
$\Rightarrow a^2 = 10a \left( \because b^2 = 5a \right)$
$\Rightarrow a = 10$
$\Rightarrow b^2 = 5a$
$\Rightarrow b^2 = 50$
$\text{ Substituting the values ofaandbin the equation of an ellipse, we get }:$
$\frac{x^2}{100} + \frac{y^2}{50} = 1$
$\therefore x^2 + 2 y^2 = 100$
$\text{This is the required equation of the ellipse }.$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 26 Ellipse
Exercise 26.1 | Q 7 | Page 23