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

Answer 1

Answer: (D)

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