A planet is revolving round the sun is elliptical orbit. Velocity at perigee position ( nearest) is `v_(1)` and at apogee position ( farthest) is `v_(2)`. Both these velocities are perpendicular to the line joining centre of sun and planet. `r_(1)` is the minimum distance and `r_(2)` the maximum distance.
At apogee position suppose speed of planet is slightly decreased from `v_(2)`, then what will happen to minimum distance `r_(1)` and maximum distance `r_(2)` in the subsequent motion.

Options

A. `r_(1)` and `r_(2)` both will decrease
B. `r_(1)` and `r_(2)` both will increase
C. `r_(2)` will remain as it is while `r_(1)` will increase
D. `r_(2)` will remain as it is while `r_(1)` will decrease

Answer

To solve the problem, we need to analyze the motion of a planet in an elliptical orbit around the Sun, particularly focusing on the effects of a decrease in velocity at the apogee position.

### Step-by-Step Solution:

1. **Understanding the Orbit**:
- A planet in an elliptical orbit has two key positions: perigee (closest point to the Sun) and apogee (farthest point from the Sun).
- Let \( r_1 \) be the distance at perigee and \( r_2 \) be the distance at apogee.

2. **Initial Conditions**:
- The planet has velocities \( v_1 \) at perigee and \( v_2 \) at apogee, both perpendicular to the line joining the Sun and the planet.
- According to Kepler's laws, the area swept out by the line joining the planet and the Sun is constant.

3. **Effect of Decreasing Velocity at Apogee**:
- When the speed of the planet at apogee is slightly decreased from \( v_2 \), the total mechanical energy of the planet in its orbit decreases.
- This decrease in energy causes the orbit to become more elliptical.

4. **Conservation of Angular Momentum**:
- The angular momentum of the planet must be conserved. Since the velocity at apogee is decreased, the distance \( r_2 \) must remain constant to conserve angular momentum.
- Therefore, \( r_2 \) remains unchanged.

5. **Change in Perigee Distance**:
- As the orbit becomes more elliptical, the minimum distance \( r_1 \) will decrease. This is because the planet will move closer to the Sun at perigee after the decrease in velocity at apogee.
- Thus, \( r_1 \) decreases while \( r_2 \) remains constant.

### Conclusion:
- After the decrease in speed at apogee, the minimum distance \( r_1 \) will decrease, while the maximum distance \( r_2 \) will remain unchanged.