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 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 $\frac{8}{3}$

Given directrix is $x=4$

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

Focus is at $(0,0)$

Shifting focus to $(h,0)$ such that centre becomes $(0,0).$

Equation of directrix becomes $x=4+h$ which gives

Distance between center and focus i.e, $ae=h$ and

Distance between center and directrix i.e, $\frac{a}{e}=4+h$

Solving for $a$

$e=\frac{1}{2}\Rightarrow a=2h$ and $2a=4+h$

$2a=4+\frac{a}{2}\Rightarrow a=\frac{8}{3}.$