1. Identify where minimum and maximum speeds occur

For a planet moving in an elliptical orbit around the sun:

• Minimum distance from sun $= x_1$ (perihelion)
• Maximum distance from sun $= x_2$ (aphelion)

The planet moves:

• fastest at perihelion
• slowest at aphelion

So given that the minimum speed is $v_0$, this must be the speed at distance $x_2$.

Thus, $v_{\min} = v_0 \quad \text{at} \quad r=x_2$

We need the maximum speed $v_{\max}$ at $r=x_1$.

2. Use conservation of angular momentum

For motion under central force, angular momentum is conserved: $mrv_\perp = \text{constant}$

At perihelion and aphelion, velocity is perpendicular to radius vector, so: $m x_1 v_{\max} = m x_2 v_0$

Cancel $m$: $x_1 v_{\max} = x_2 v_0$

Hence, $v_{\max} = \frac{v_0 x_2}{x_1}$

3. Match with options

$v_{\max} = \frac{v_0 x_2}{x_1}$

This corresponds to Option D.

4. Verification with stored answer

Stored correct answer: D

Our derived answer: D

So they agree.