1. Use Kepler's third law

For a planet revolving around the Sun, $T^2 \propto r^3$ where $T$ is the time period and $r$ is the orbital radius.

So, $T \propto r^{3/2}$

2. Given initial values

Initial time period: $T_1 = 200\text{ days}$

New distance is one fourth of the original distance: $r_2 = \frac{r_1}{4}$

3. Relate the two time periods

Using $\frac{T_2}{T_1} = \left(\frac{r_2}{r_1}\right)^{3/2}$ we get $\frac{T_2}{200} = \left(\frac{1}{4}\right)^{3/2}$

Now, $\left(\frac{1}{4}\right)^{3/2} = \left(\frac{1}{2^2}\right)^{3/2} = \frac{1}{2^3} = \frac{1}{8}$

Therefore, $T_2 = 200 \times \frac{1}{8} = 25\text{ days}$

4. Check options

• A: $20$ ❌
• B: $50$ ❌
• C: $100$ ❌
• D: $25$ ✅

So the correct answer is D.