Let the equations of two ellipses be E_1:x^23+y^22=1 and E_2:x^216+y^2b^2=1.

Question

Let the equations of two ellipses be $E_1:\dfrac{x^2}{3}+\dfrac{y^2}{2}=1$ and $E_2:\dfrac{x^2}{16}+\dfrac{y^2}{b^2}=1$. If the product of their eccentricities is $	frac12$, then the length of the minor axis of ellipse $E_2$ is:

Accepted Answer

$e_1=\sqrt{1-	frac23}=\dfrac{1}{\sqrt3}$. Given $e_1e_2=	frac12\Rightarrow e_2=\dfrac{\sqrt3}{2}$.

$e_2^2=1-\dfrac{b^2}{16}=\dfrac34\Rightarrow \dfrac{b^2}{16}=\dfrac14\Rightarrow b^2=4\Rightarrow b=2$.

Minor axis length $=2b=4.$

Solutions · 1