\(K.E\) of a planet \(=\frac{1}{2} m v^2\)

\(K.E\) at perigee \(=\frac{1}{2} m v_P^2\)

\(K.E\) at apogee \(=\frac{1}{2} m v_A^2\)

Using conservation of angular momentum at \(P\) and \(A\)

\(\Rightarrow m v_P r_P=m v_A r_A\)

\(\Rightarrow \frac{v_P}{v_A}=\frac{r_A}{r_P}=\frac{a(1+e)}{a(1-e)}\)

\(\Rightarrow \frac{ K \cdot E _P}{ K \cdot E _A}=\frac{v_P^2}{v_A^2}=\left(\frac{1+e}{1-e}\right)^2\)