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  • The eccentricity of an ellipse with centre at the origin and axes along the coordinate axes, is . If one of the di

The eccentricity of an ellipse with centre at the origin and axes along the coordinate axes, is 1 / 2. If one of the directrices is \mathrm{x=4}, then the equation of the ellipse is

Option: 1

\mathrm{4 x^2+3 y^2=1}


Option: 2

\mathrm{3 x^2+4 y^2=12}


Option: 3

\mathrm{4 x^2+3 y^2=12}


Option: 4

\mathrm{3 x^2+4 y^2=1}


Answers (1)

best_answer

Let the equation of the ellipse be  \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \text {. }}

We have, \mathrm{e=\frac{1}{2}} and \mathrm{\frac{a}{e}=4}

\mathrm{ \begin{aligned} & \Rightarrow \quad a=2 \\ & \Rightarrow b^2=2^2\left(1-\frac{1}{4}\right) \quad\left[\because b^2=a^2\left(1-e^2\right)\right] \\ & \Rightarrow b^2=3 \end{aligned} }
Hence, the equation of the ellipse is \mathrm{\frac{x^2}{4}+\frac{y^2}{3}=1}

\mathrm{ \Rightarrow 3 x^2+4 y^2=12 }

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