According to the principle of conservation of angular momentum, the angular momentum of the planet remains constant throughout its orbit. At the positions of closest approach (perihelion) and farthest distance (aphelion), the velocity vector is perpendicular to the radius vector. Therefore, the angular momentum is given by $L = mvr$. Equating the angular momentum at both points: $m v_1 r_1 = m v_2 r_2$ Dividing both sides by $m v_2 r_1$: $\frac{v_1}{v_2} = \frac{r_2}{r_1}$.