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The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is $x=4$ , then the equation of the ellipse is :

Option 1)
$4x^{2}+3y^{2}=12$

Option 2)
$3x^{2}+4y^{2}=12$

Option 3)
$3x^{2}+4y^{2}=1$

Option 4)
$4x^{2}+3y^{2}=1$

Answers (1)

As we learnt in

Standard equation -

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1$

- wherein

$a\rightarrow$ Semi major axis

$b\rightarrow$ Semi minor axis

Eccentricity -

$e= \sqrt{1-\frac{b^{2}}{a^{2}}}$

- wherein

For the ellipse

$\frac{x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}= 1$

Equation of directrices -

$x= \pm \frac{a}{e}$

- wherein

For the ellipse

$\frac{x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}= 1$

$e=\frac{1}{2} \:and\: \frac{a}{e}=4$

$Thus\: \:\frac{a}{\frac{1}{2}}=4$

$=>a=2$

$Also \:b^{2}=a^{2}\left ( 1-e^{2} \right )$ $=4\left ( 1-\frac{1}{4} \right )$ $=3$

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

$=> 3x^{2}+4y^{2}=12$

Option 1)

$4x^{2}+3y^{2}=12$

Incorrect option

Option 2)

$3x^{2}+4y^{2}=12$

Correct option

Option 3)

$3x^{2}+4y^{2}=1$

Incorrect option

Option 4)

$4x^{2}+3y^{2}=1$

Incorrect option

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