If the distance between the foci of an ellipse is half the length of its latus rectum, the…

Question

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:

Accepted Answer

Distance between foci $=2ae$; latus rectum $=\dfrac{2b^2}{a}$. Given $2ae=\dfrac12\cdot\dfrac{2b^2}{a}=\dfrac{b^2}{a}$:
$$2a^2e=b^2=a^2(1-e^2)\ \Rightarrow\ 2e=1-e^2\ \Rightarrow\ e^2+2e-1=0\ \Rightarrow\ e=\sqrt2-1.$$

Solutions · 1