If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentri…

Length of minor axis \(=2 b=\frac{1}{4}(2 a e)\)

\(\Rightarrow 4b=ae\Rightarrow 16{b}^{2}={a}^{2}{e}^{2}\)

\(\Rightarrow 16\left({a}^{2}-{a}^{2}{e}^{2}\right)={a}^{2}{e}^{2}\Rightarrow 16{a}^{2}=17{a}^{2}{e}^{2}\)

\(\Rightarrow {e}^{2}=\frac{16}{17}\Rightarrow e=\frac{4}{\sqrt{17}}\)