To find the equation of the ellipse for the satellite's orbit, we start by using the information provided:

• The distance of the satellite from the planet varies from 300,000 km to 500,000 km.
• The vertices are on the x-axis.
• The center of the ellipse is at the origin.
• The units are in $100,000$ km.

Step 1: Convert the distances into units

Given that the distances are in $100,000$ km, we convert:

• The minimum distance (closest distance): $300,000 \text{ km} = 3$ units.
• The maximum distance (farthest distance): $500,000 \text{ km} = 5$ units.

Step 2: Determine the values of $a$, $c$, and $b$

In an ellipse centered at the origin, with the major axis along the x-axis, the standard form of the equation is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Where:

• $a$ is the semi-major axis.
• $c$ is the distance from the center to each focus.
• $b$ is the semi-minor axis.

Vertices: The vertices are located at $(\pm a, 0)$, and since the vertices are at $\pm 5$ units: $a = 5$

Foci: The planet is at one of the foci of the ellipse. The distance from the center to the focus is $c$, and this focus lies on the x-axis. The closest distance to the planet (at the other focus) is the periapsis (3 units). So, the distance to the focus $c$ is the difference between the semi-major axis and the periapsis: $c = a - \text{periapsis} = 5 - 3 = 2$

Calculate $b^2$ using the relationship $c^2 = a^2 - b^2$: $c^2 = a^2 - b^2 \Rightarrow 2^2 = 5^2 - b^2$ $4 = 25 - b^2$ $b^2 = 21$

Step 3: Write the equation of the ellipse

Now, substitute $a^2 = 25$ and $b^2 = 21$ into the standard form equation: $\frac{x^2}{25} + \frac{y^2}{21} = 1$

This is the equation of the ellipse for the satellite's orbit.

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Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the significance of the semi-minor axis in the context of an orbit?
2. How does the eccentricity of the ellipse relate to the shape of the orbit?
3. What would the equation of the ellipse be if the planet was not at the origin but at a different point?
4. How would the equation change if the major axis was along the y-axis instead?
5. What happens to the orbit if the distance from the satellite to the planet increases uniformly?

Tip: Remember that the foci of an ellipse play a crucial role in defining the eccentricity, which determines how "stretched" the ellipse is. The closer the foci are to each other, the more circular the ellipse.