The eccentricity of an ellipse centered at th | vedclass

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(B) The equation of the directrix is $x = 4$,which is parallel to the $y$-axis. This implies the major axis of the ellipse lies along the $x$-axis.Giv

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

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(B) The equation of the directrix is $x = 4$,which is parallel to the $y$-axis. This implies the major axis of the ellipse lies along the $x$-axis.Given the center is at the origin $(0, 0)$,the equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$.The distance of the directrix from the center is given by $\frac{a}{e} = 4$.Given $e = 1/2$,we have $\frac{a}{1/2} = 4$,which implies $a = 2$.Using the relation $b^2 = a^2(1 - e^2)$,we get $b^2 = 2^2(1 - (1/2)^2) = 4(1 - 1/4) = 4(3/4) = 3$.Substituting $a^2 = 4$ and $b^2 = 3$ into the standard equation,we get $\frac{x^2}{4} + \frac{y^2}{3} = 1$.Multiplying by $12$,we obtain $3x^2 + 4y^2 = 12$.

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

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