A focus of an ellipse is at the origin.

Question

A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $	frac12$. Then the length of the semi-major axis is:

Accepted Answer

Distance from focus to directrix $=\dfrac ae-ae=4$ (standard relation using $a/e$ and $ae$ measured from the centre, with the focus at distance $ae$ inside):
$$a\left(\frac1e-e\right)=4\Rightarrow a\left(2-\frac12\right)=4\Rightarrow a\cdot\frac32=4\Rightarrow a=\frac83.$$

Solutions · 1