Question

The eccentricity of an ellipse with its centre at the origin is $\frac{1}{2}$. If one of the directrices is $x=4$, then the equation of the ellipse is

• A. $4x^2+3y^2=12$
• B. $3x^2+4y^2=12$
• C. $3x^2+4y^2=1$
• D. $4x^2+3y^2=1$

Answer 1

The correct option is B $3x^2+4y^2=12$

Equation of directrix is $x=4$ which is parallel to y-axis so major axis of the ellipse is x-axis and given center of ellipse is origin. Let equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
Given,eccentricity $e=\frac{1}{2}$ ,we know that distance of directrix from center is $\frac{a}{e}=4⟹a=2$,from eccentricity definition we get value of $b$ as $\sqrt{3}$,now equation becomes
$\frac{x^2}{4}+\frac{y^2}{3}=1⟹3x^2+4y^2=12$