W4. 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 planet 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 y-axis. Assume all units are in 100,000
km
To find the equation of the ellipse, we first need to determine the distance between the two vertices, which is the major axis of the ellipse. In this case, the distance is 500,000 km - 300,000 km = 200,000 km.
Since the vertices are on the y-axis, the major axis of the ellipse is along the y-axis, making the equation of the ellipse:
y^2/b^2 + x^2/a^2 = 1
where "a" is the semi-major axis and "b" is the semi-minor axis.
Since the distance between the two vertices is 200,000 km, the full length of the major axis is 2a = 200,000 km. Therefore, a = 100,000 km.
The distance from the center of the ellipse to each vertex is a, so the coordinates of the vertices are (0, 100,000) and (0, -100,000).
The distance from the center to a vertex is a, while the distance from the center to a co-vertex is b. In this case, the distance from the center to each co-vertex is 150,000 km, which makes b = 150,000 km.
Therefore, the equation of the ellipse is:
y^2/150,000^2 + x^2/100,000^2 = 1
This simplifies to:
y^2/225,000,000,000 + x^2/10,000,000,000 = 1
And the final equation of the ellipse is:
x^2/10,000,000,000 + y^2/225,000,000,000 = 1