The distance between the closest planet and the sun rise is r1 while the distance between the farthest planet and the sun is r2. The linear velocities of these planets is v1 and v2 respectively. What is the ratio of v1/v2

The correct option/answer is r2/r1.

Let us break down the physics behind this result step-by-step.
When a planet orbits the Sun under the influence of the gravitational force, the net external torque acting on the planet about the Sun is zero. This is because the gravitational force is a central force, always directed along the line joining the planet and the Sun.

According to the law of conservation of angular momentum, since the net external torque is zero, the orbital angular momentum ($L$) of the planet is conserved.

The magnitude of angular momentum $L$ of a planet of mass $m$ at a distance $r$ from the Sun, moving with velocity $v$ perpendicular to the position vector (which is true at the points of closest approach, perihelion, and farthest distance, aphelion), is given by:
$L=m·v·r$

Applying this conservation law to the closest planet (with distance $r_1$ and velocity $v_1$) and the farthest planet (with distance $r_2$ and velocity $v_2$), assuming both planets have comparable orbital dynamics or by analyzing the same planet at its closest and farthest points in an elliptical orbit, we get:
$m·v_1·r_1=m·v_2·r_2$

By cancelling the mass $m$ from both sides, we obtain the relation:
$v_1·r_1=v_2·r_2$

Rearranging this equation to find the ratio of their linear velocities ($v_1/v_2$):
$\frac{v_1}{v_2}=\frac{r_2}{r_1}$

Hence, the ratio of the linear velocities of the planets is inversely proportional to the ratio of their distances, which is r2/r1.