Given that the orbit of the satellite is an ellipse with the planet at one focus and the vertices on the r-axis, we can find the equation of the ellipse.

Step 1: Understand the ellipse parameters

• The center of the ellipse is at the origin $(0, 0)$.
• The vertices of the ellipse are on the r-axis (which we'll consider as the x-axis).

Given:

• The distance from the satellite to the planet varies from 300,000 km to 500,000 km.
• The distance is measured along the major axis, from the center of the ellipse to the vertices.

Step 2: Assign the values to the ellipse parameters

Since the units are in $100,000 \text{ km}$, let's express the distances as:

• Minimum distance (closest to the planet): $3$ units ($300,000$ km).
• Maximum distance (farthest from the planet): $5$ units ($500,000$ km).

These distances correspond