Latus rectum of an ellipse is 10 and minor axis is equal to distance between foci. equation of ellip

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10-2. Parabola, Ellipse, Hyperbola

easy

The latus rectum of an ellipse is $10$ and the minor axis is equal to the distance between the foci. The equation of the ellipse is

A

${x^2} + 2{y^2} = 100$

B

${x^2} + \sqrt 2 {y^2} = 10$

C

${x^2} - 2{y^2} = 100$

D

None of these

Solution

(a) Given $\frac{{2{b^2}}}{a} = 10$ and $2b = 2ae$

Also ${b^2} = {a^2}(1 – {e^2})$

${e^2} = (1 – {e^2})$

$e = \frac{1}{{\sqrt 2 }}$

$b = \frac{a}{{\sqrt 2 }}$ or $b = 5\sqrt 2 $, $a = 10$

Hence equation of ellipse is $\frac{{{x^2}}}{{{{(10)}^2}}} + \frac{{{y^2}}}{{{{(5\sqrt 2 )}^2}}} = 1$

$i.e.$, ${x^2} + 2{y^2} = 100$.

Standard 11

Mathematics

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