Question
A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the 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 :
$$\frac{8}{3}$$
Solution :
Perpendicular distance of directrix from focus
$$\eqalign{ & = \frac{a}{e} - ae = 4 \cr & \Rightarrow a\left( {2 - \frac{1}{2}} \right) = 4 \cr & \Rightarrow a = \frac{8}{3} \cr} $$
$$\therefore $$ Semi major axis $$ = \frac{8}{3}$$
Perpendicular distance of directrix from focus
$$\eqalign{ & = \frac{a}{e} - ae = 4 \cr & \Rightarrow a\left( {2 - \frac{1}{2}} \right) = 4 \cr & \Rightarrow a = \frac{8}{3} \cr} $$
$$\therefore $$ Semi major axis $$ = \frac{8}{3}$$