The eccentricity of an ellipse, with its centre as origin, is \(1 / 2\). If one of the directrices is…
The eccentricity of an ellipse, with its centre as origin, is \(1 / 2\). If one of the directrices is \(x=4\), then the equation of the ellipse is given by
\(4 x^2+y^2=12\)
\(x^2+3 y^2=12\)
\(4 x^2+3 y^2=12\)
\(3 x^2+4 y^2=12\)
Solution
Centre \(=(0,0)\)
Eccentricity \((\mathrm{e})=\frac{1}{2}\)
Equation of directrix is \(x=4\)
\(\Rightarrow \frac{a}{e}=4\)
\(\Rightarrow a=4 e\)
\(\Rightarrow a=2\)
\(\begin{aligned}
b^2 & =a^2\left(1-e^2\right) \\
& =4\left(1-\frac{1}{4}\right) \\
b^2 & =3
\end{aligned}\)
\(\therefore\) Required Equation of Ellipse is \(\frac{x^2}{4}+\frac{y^2}{3}=1\)
\(3 x^2+4 y^2=12\)
Hence, option (d) is correct.