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.
1
To find the equation of the ellipse, we first need to find the coordinates of the vertices. Since the vertices of the ellipse lie on the y-axis, the coordinates of the vertices are (0, 3) and (0, 5).
The distance from the center of the ellipse to the vertices is also known as the major axis, which is the distance between the two vertices of the ellipse.
The formula for the major axis of an ellipse with the center at the origin and vertices on the y-axis is given by:
a = distance from center to vertex 1 = distance from center to vertex 2 = 5 - 3 = 2
The minor axis of an ellipse is the distance between the two co-vertices. Since the ellipse is vertically oriented, the minor axis is the same as the distance between the two co-vertices.
The co-vertices are halfway between the center and each vertex. Therefore, the distance from the center to the co-vertices is:
b = (3 + 5) / 2 = 4
The equation of an ellipse centered at the origin is:
x^2 / a^2 + y^2 / b^2 = 1
Plugging in the values of a and b, we have:
x^2 / 2^2 + y^2 / 4^2 = 1
x^2 / 4 + y^2 / 16 = 1
Thus, the equation of the ellipse is:
x^2 / 4 + y^2 / 16 = 1