The eccentricity of an ellipse,with its centr | vedclass

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(D) Given: Centre is $(0,0)$,eccentricity $e = 1/2$,and directrix $x = 4$.For an ellipse with centre at origin and directrix $x = a/e$,we have $a/e =

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

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(D) Given: Centre is $(0,0)$,eccentricity $e = 1/2$,and directrix $x = 4$.For an ellipse with centre at origin and directrix $x = a/e$,we have $a/e = 4$.Substituting $e = 1/2$,we get $a = 4 	imes (1/2) = 2$.Now,$b^2 = a^2(1 - e^2) = 4(1 - 1/4) = 4(3/4) = 3$.The standard equation of the ellipse 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 $3x^2 + 4y^2 = 12$.Thus,option $D$ is correct.

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

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