For an ellipse with eccentricity $\frac{1}{2}$ the centre is at the origin. If one directrix is $x=4$, then…
For an ellipse with eccentricity $\frac{1}{2}$ the centre is at the origin. If one directrix is $x=4$, then the equation of the ellipse is
- $3 x^2+4 y^2=1$
- $3 x^2+4 y^2=12$
- $4 x^2+3 y^2=1$
- $4 x^2+3 y^2=12$
Solution
Given that, $e=\frac{1}{2}$ and $\frac{a}{e}=4$
$
\begin{array}{lrl}
\Rightarrow & \frac{a}{1 / 2}=4 \Rightarrow a=2 \\
\text { Using } & b^2=a^2\left(1-e^2\right) \\
\Rightarrow & b^2=4\left(1-\frac{1}{4}\right)=3
\end{array}
$
$\therefore$ Equation of ellipse is
$
\begin{aligned}
\frac{x^2}{4}+\frac{y^2}{3} & =1 \\
\Rightarrow \quad 3 x^2+4 y^2 & =12
\end{aligned}
$
Asked in: AP EAMCET 2008
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