Question

A planet is revolving around the sun in an elliptical orbit having eccentricity $(e = \frac{\pi}{4})$. If the period of revolution is 16 months then find the ratio of time takes by the planet going from $A$ to $B$ and from $C$ to $A$

Answer 1

$\frac{4+\pi}{4-\pi}$

Answer 2

According to Kepler's second law, the time taken to traverse a certain part of the orbit is proportional to the area swept by the radius vector from the sun to the planet. The ratio of times $\frac{t_{AB}}{t_{CA}}$ is equal to the ratio of the areas swept, which simplifies to $\frac{1+e}{1-e}$. Substituting $e = \frac{\pi}{4}$, we get $\frac{1 + \frac{\pi}{4}}{1 - \frac{\pi}{4}} = \frac{4+\pi}{4-\pi}$. The period of revolution is not needed for this ratio.