Question

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

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

Answer 1

The correct option is A $\frac{8}{3}$

Focus$=(0,0)S$

Line$:x=4$,

we know, $SP=ePM$

$\Rightarrow (x-0{)}^2+(y-0{)}^2=\frac{(x+4{)}^2}{1+0}{\left(\frac{1}{2}\right)}^2$

$\Rightarrow \frac{3x^2}{4}+\frac{8x}{4}-\frac{16}{4}+y^2=0$

$\Rightarrow \frac{3}{4}(x+\frac{4}{3}{)}^2-\frac{16}{3}+y^2=0$

$\Rightarrow \frac{3}{4}(x+\frac{4}{3}{)}^2+y^2=\frac{16}{3}$

$\Rightarrow \frac{(x+\frac{4}{3}{)}^2}{\frac{64}{9}}+\frac{y^2}{\frac{16}{3}}=1$

Now, $a^2=\frac{64}{9}$

$\Rightarrow a=\frac{8}{3}$ (Semi-major axis$\frac{1}{2}\left(2a\right)=a$)