Question

The focus of an ellipse is at the origin. The directrix is the line $x=4$ and its eccentricity is $\frac{1}{2}$ then the length of its semi-major axis is

• A. $\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $\frac{5}{3}$
• D. $\frac{8}{3}$

Answer 1

The correct option is D

$\frac{8}{3}$

Directrix is $x=4$

Eccentricity $=\frac{1}{2}$

Directrix $=[\frac{a}{e}–ae]=4$

$$\begin{gathered} \Rightarrow \frac{a}{e}–ae=4 \\ \Rightarrow a[\frac{1}{e}–e]=4 \\ \Rightarrow a[\frac{1}{2}-2]=4 \\ \Rightarrow a\times \frac{3}{2}=4 \\ \therefore a=\frac{8}{3} \end{gathered}$$

Hence, the length of its semi-major axis is $\frac{8}{3}$

Hence, the correct option is D.