Question

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

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

Answer 1

The correct option is B

$\frac{8}{3}$

Step1. Define the formula of directrix.

Given major axis is along the $\mathrm{x}-\mathrm{axis}$

Focus is origin

As per the diagram

$\begin{array}{rcl}\mathrm{CD}& =& \frac{\mathrm{a}}{\mathrm{e}}\\ \mathrm{CF}& =& \mathrm{ae}\\ \mathrm{FD}& =& 4\end{array}$

Distance between focus and directrix is $\mathrm{FD}$

$\mathrm{FD}=\mathrm{CD}-\mathrm{CF}$

Step2. Calculate the value of the length of the semi-major axis.

Directrix is the line

$\mathrm{x}=4$

Eccentricity

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

$\begin{array}{rcl}4& =& \left|\frac{\mathrm{a}}{\mathrm{e}}-\mathrm{ae}\right|\left[Substitutethevalueintheformulaweget\right]\\ & \Rightarrow & 4=\mathrm{a}\left|\frac{1}{\mathrm{e}}-\mathrm{e}\right|\\ & \Rightarrow & 4=\mathrm{a}\left|\frac{1}{{\displaystyle \frac{1}{2}}}-\frac{1}{2}\right|\\ & \Rightarrow & 4=\mathrm{a}\left|2-\frac{1}{2}\right|\\ & \Rightarrow & 4=\mathrm{a}\left(\frac{3}{2}\right)\\ & \Rightarrow & \mathbf{a}\mathbf{=}\frac{\mathbf{8}}{\mathbf{3}}\end{array}$

Hence, Option (B) is the correct answer.