An orbit of a satellite around a planet is an ellipse, with the planet at one focus of this ellipse. The distance of the satellite from this star varies from 300.000 km to 500,000 km, attained when the satellite is at each of the two vertices. Find the equation of this ellipse, if its center is at the origin, and the vertices are on the r-axis. Assume all units are in 100.000 km.

2. The orbit of a planet has the shape of an ellipse, and on one of the foci is the star around which it revolves. The planet is closest to the star when it is at one vertex. It is farthest from the star when it is at the other vertex. Suppose the closest and farthest distances of the planet from this star, are 420 million kilometers and 580 million kilometers, respectively. Find the equation of the ellipse, in standard form, with center at the origin and the star at the x-axis. Assume all units are in millions of kilometers.

To solve these problems, we'll use the properties of ellipses. An ellipse with its center at the origin and aligned with the coordinate axes can be described by the equation:

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

where \( a \) is the semi-major axis (half of the longest diameter) and \( b \) is the semi-minor axis (half of the shortest diameter). The foci are located at \( \pm c \) along the major axis, where \( c = \sqrt{a^2 - b^2} \).

### Problem 1: Satellite Orbit
- **Closest distance to the planet** (periapsis): 300,000 km
- **Farthest distance from the planet** (apoapsis): 500,000 km

The distance from the center of the ellipse to the vertices is the semi-major axis \( a \), and the total distance is 800,000 km. Hence:

\[
a = \frac{500,000 + 300,000}{2} = 400,000 \, \text{km}
\]

The distance between the foci (which is twice the distance from the center to a focus, \( 2c \)) is the difference between the apoapsis and periapsis distances:

\[
2c = 500,000 - 300,000 = 200,000 \, \text{km}
\]

Thus,

\[
c = \frac{200,000}{2} = 100,000 \, \text{km}
\]

Now, using the relationship \( c = \sqrt{a^2 - b^2} \):

\[
100,000 = \sqrt{400,000^2 - b^2}
\]

\[
(100,000)^2 = (400,000)^2 - b^2
\]

\[
10^8 = 16 \times 10^8 - b^2
\]

\[
b^2 = 16 \times 10^8 - 10^8 = 15 \times 10^8
\]

\[
b = \sqrt{15} \times 10^4 \approx 387,298 \, \text{km}
\]

Since units are in 100,000 km:

- \( a = 4 \)
- \( b = \sqrt{15} \approx 3.87 \)

The equation of the ellipse is:

\[
\frac{x^2}{4^2} + \frac{y^2}{(\sqrt{15})^2} = 1 \implies \frac{x^2}{16} + \frac{y^2}{15} = 1
\]

### Problem 2: Planet Orbit
- **Closest distance to the star**: 420 million km
- **Farthest distance from the star**: 580 million km

The total length of the major axis is:

\[
a = \frac{580 + 420}{2} = 500 \, \text{million km}
\]

The distance between the foci (distance from closest to farthest point) is:

\[
2c = 580 - 420 = 160 \, \text{million km}
\]

So,

\[
c = \frac{160}{2} = 80 \, \text{million km}
\]

Now using \( c = \sqrt{a^2 - b^2} \):

\[
80 = \sqrt{500^2 - b^2}
\]

\[
80^2 = 500^2 - b^2
\]

\[
6400 = 250000 - b^2
\]

\[
b^2 = 250000 - 6400 = 243600
\]

\[
b = \sqrt{243600} \approx 493.56 \, \text{million km}
\]

The equation of the ellipse is:

\[
\frac{x^2}{500^2} + \frac{y^2}{493.56^2} = 1 \implies \frac{x^2}{250000} + \frac{y^2}{243600} = 1
\]

### Questions to Explore Further:
1. How does changing the distance between the foci affect the shape of an ellipse?
2. What is the significance of the semi-major and semi-minor axes in defining an orbit?
3. Can the equation of an ellipse be derived if only one of the distances (closest or farthest) is known?
4. How are ellipses used to model other celestial orbits, such as those of comets?
5. How does the eccentricity of an ellipse relate to the orbit's shape and the distance between foci?

### Tip:
In problems involving orbits, distances from the central focus to the orbiting object at closest and farthest points provide direct information about the lengths of the semi-major axis and the focal distance, which are key to determining the equation of the orbit.

Discover how to find the equation of an ellipse describing the orbits of satellites and planets. Includes step-by-step solutions for determining semi-major axes, focal distances, and ellipse equations. Suitable for high school students and enthusiasts of celestial mechanics.

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