The eccentricity of an ellipse,with its centr | vedclass

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(B) Given that the directrix is $x = 4$,which is parallel to the $y$-axis,the major axis of the ellipse lies along the $x$-axis.Let the equation of th

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

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(B) Given that the directrix is $x = 4$,which is parallel to the $y$-axis,the major axis of the ellipse lies along the $x$-axis.Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,where $a > b$.The eccentricity $e$ is given as $\frac{1}{2}$.Using the relation $e^2 = 1 - \frac{b^2}{a^2}$,we have $\frac{b^2}{a^2} = 1 - e^2 = 1 - (\frac{1}{2})^2 = 1 - \frac{1}{4} = \frac{3}{4}$.The equation of the directrix is $x = \frac{a}{e}$. Given $x = 4$,we have $\frac{a}{e} = 4$.Substituting $e = \frac{1}{2}$,we get $a = 4e = 4 	imes \frac{1}{2} = 2$.Now,$b^2 = \frac{3}{4}a^2 = \frac{3}{4} 	imes (2)^2 = \frac{3}{4} 	imes 4 = 3$.Substituting $a^2 = 4$ and $b^2 = 3$ into the ellipse equation,we get $\frac{x^2}{4} + \frac{y^2}{3} = 1$.Multiplying by $12$,we get $3x^2 + 4y^2 = 12$.

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

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