Question

The eccentricity of an ellipse with it's centre at origin is $\frac{1}{2}$. If one of the directrices is $x=8$, then the equation of the ellipse is given by

• A. $3x^2+4y^2=48$
• B. $3x^2+4y^2=36$
• C. $4x^2+3y^2=36$
• D. $4x^2+3y^2=48$

Answer 1

The correct option is A $3x^2+4y^2=48$

Since, directrix is parallel to y-axis and center is origin, axis of ellipse is $x$-axis.
Let ellipse be represented by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$
Then, $e^2=1-\frac{b^2}{a^2}$
$\frac{b^2}{a^2}=1-e^2=\frac{3}{4}$
Also, one directrix is $x=8$
$\frac{a}{e}=8\Rightarrow a=4$
$b^2=3\times 4=12$
Equation of ellipse becomes
$\frac{x^2}{16}+\frac{y^2}{12}=1$
$3x^2+4y^2=48$