For an ellipse with eccentricity 1/2 and  centre is at origin. If one directrix is x = 4 then the equation of ellipse is

Given $\mathrm{e}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

and directrix is  $\mathrm{x}=\frac{\mathrm{a}}{\mathrm{e}}=4$

$$\begin{gathered} \Rightarrow a=4e \\ =2 \end{gathered}$$

$$\begin{gathered} \mathrm{Now} {\mathrm{b}}^2={\mathrm{a}}^2(1-{\mathrm{e}}^2) \\ =4\left(1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right) \\ =3 \end{gathered}$$

$\therefore$equation of an ellipse is 

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