Explanation:

Given:
- A planet is revolving around a star in an elliptic orbit.
- The ratio of the farthest distance to the closest distance of the planet from the star is 4.

Kepler's Second Law:
Kepler's Second Law states that the line joining a planet to the star sweeps out equal areas in equal times. This means that the speed of the planet is not constant throughout its orbit. As the planet gets closer to the star, it moves faster, and as it gets farther away, it slows down.

Kinetic Energy:
The kinetic energy of an object is given by the formula KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

Analysis:
1. Let's assume that the closest distance of the planet from the star is d.
2. According to the given information, the farthest distance of the planet from the star is 4d.
3. As the planet moves from the closest distance to the farthest distance, its speed decreases.
4. Since kinetic energy is directly proportional to the square of velocity, the kinetic energy of the planet at the farthest distance will be less than its kinetic energy at the closest distance.

Calculating the Ratio of Kinetic Energies:
1. Let's assume the mass of the planet remains constant throughout its orbit.
2. At the closest distance, let's assume the velocity of the planet is v1. Therefore, the kinetic energy at the closest distance is KE1 = 1/2 * m * v1^2.
3. At the farthest distance, let's assume the velocity of the planet is v2. Therefore, the kinetic energy at the farthest distance is KE2 = 1/2 * m * v2^2.
4. According to Kepler's Second Law, the planet sweeps out equal areas in equal times. This means that the time taken by the planet to cover the closest distance is equal to the time taken to cover the farthest distance.
5. Since the mass of the planet remains constant, we can write v1 * t1 = v2 * t2, where t1 and t2 are the times taken to cover the closest and farthest distances, respectively.
6. Since the time taken is the same, we can write v1/v2 = t2/t1.
7. As the planet gets farther away, it slows down, so v2 < />
8. Therefore, t2/t1 < />
9. Since v1/v2 = t2/t1, we can say that v1/v2 < />
10. Since the kinetic energy is directly proportional to the square of velocity, (v1/v2)^2 < />
11. Therefore, KE1/KE2 = (v1^2)/(v2^2) < />
12. We know that the ratio of the farthest distance to the closest distance is 4. Therefore, v1/v2 = 4.
13. Substituting this value in the equation, (4^2) = KE1/KE2.
14. Simplifying, we get 16 = KE1/KE2.

Conclusion:
The ratio of the kinetic energies of the planet at the closest to farthest