A focus of an ellipse is at the origin. The directrix is the line  $x=4$ and the eccentricity is ½. Then the length of the semi–major axis is

Obviously the major axis is along the x-axis
The distance between the focus and the corresponding 
Directrix  $=|\frac{a}{e}-ae|=4$
 $$\begin{gathered} \Rightarrow |\frac{a}{e}-ae|=4(notethat\frac{a}{e}>ae) \\ \Rightarrow a(\frac{1}{e}-e)=4\to a(2-\frac{1}{2})=4 \\ \Rightarrow a.\frac{3}{2}=4\therefore a=\frac{8}{3} \end{gathered}$$