For an ellipse with eccentricity e = frac{1}{ | vedclass

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(A) Given,eccentricity $e = \frac{1}{2}$,centre is $(0, 0)$,and the equation of the directrix is $x = \frac{a}{e} = 4$.Since $e = \frac{1}{2}$,we have

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

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(A) Given,eccentricity $e = \frac{1}{2}$,centre is $(0, 0)$,and the equation of the directrix is $x = \frac{a}{e} = 4$.Since $e = \frac{1}{2}$,we have $\frac{a}{1/2} = 4$,which implies $a = 2$.Now,$b^2 = a^2(1 - e^2) = 2^2(1 - (\frac{1}{2})^2) = 4(1 - \frac{1}{4}) = 4(\frac{3}{4}) = 3$.The equation of the ellipse with centre at the origin and major axis along the $x$-axis is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.Substituting the values,we get $\frac{x^2}{4} + \frac{y^2}{3} = 1$.Multiplying by $12$,we get $3x^2 + 4y^2 = 12$.

For an ellipse with eccentricity $e = \frac{1}{2}$,the centre is at the origin. If one of its directrices is $x = 4$,then the equation of the ellipse is

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