Question

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

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

Answer 1

Answer: (D)

Centre $=(0,0)$
Eccentricity $(e)=\frac{1}{2}$
Equation of directrix is $x=4$
$\Rightarrow \frac{a}{e}=4$
$\Rightarrow a=4e$
$\Rightarrow a=2$
$b^2=a^2\left(1-e^2\right)$
$=4\left(1-\frac{1}{4}\right)$
$b^2=3$
$\therefore$ Required Equation of Ellipse is
$\frac{x^2}{4}+\frac{y^2}{3}=1$
$3x^2+4y^2=12$