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 ellipse is

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

Answer 1

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

Since the directrix is $x=4$, therefore the major axis of ellipse is parallel to x-axis
$\therefore \frac{a}{e}=4$
$\Rightarrow a=4\times \frac{1}{2}$
$\Rightarrow a=2$
Also, we know that
$b^2=a^2(1-e^2)$
$\Rightarrow b^2=4(1-\frac{1}{4})=4\times \frac{3}{4}$
$\Rightarrow b^2=3$
$\therefore$ Equation of ellipse is $\frac{x^2}{4}+\frac{y^2}{3}=1$
$\Rightarrow 3x^2+4y^2=12$