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. $3x^2+4y^2=1$
• 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 $3x^2+4y^2=12$

Center of ellipse is at origin. Eccentricity $e=\frac{1}{2}$

Directrix is $x=4$

Since Directrix is parallel to y axis we know that the ellipse is a horizontal ellipse where $a>b$ in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Now $e=\sqrt{1-\frac{b^2}{a^2}}\Rightarrow \frac{1}{2}=\sqrt{1-\frac{b^2}{a^2}}\Rightarrow \frac{b^2}{a^2}=\frac{3}{4}$.......1)

Now directrix $=\frac{a}{e}=4\Rightarrow a=4e=2$

Put in 1) to get $b=\sqrt{3}$

Thus equation of ellipse is $\frac{x^2}{4}+\frac{y^2}{3}=1\Rightarrow 3x^2+4y^2=12$