Find an equation that models the paths of a satellite if its path is a hyperbola, a=45,000 km, and c=71,000 km. Assume that the center of the hyperbola is the origin and the transverse axis is horizontal.

Answer 1

Since the hyperbola is centered at the origin and the transverse axis is horizontal, the equation has the form:

x^2/a^2 – y^2/b^2 = 1

where a is the distance from the origin to either vertex, and b is the distance from the origin to either foci. We are given that a = 45,000 km, so:

x^2/45,000^2 – y^2/b^2 = 1

To find b, we use the relationship c^2 = a^2 + b^2, where c is the distance from the origin to either focus. We are given that c = 71,000 km, so:

71,000^2 = 45,000^2 + b^2

b^2 = 71,000^2 – 45,000^2

b^2 = 2,696,000,000 km^2

Now we can substitute this value of b^2 into the equation:

x^2/45,000^2 – y^2/2,696,000,000 = 1

This equation models the paths of the satellite if its path is a hyperbola, with a distance of 45,000 km from the origin to either vertex and 71,000 km from the origin to either focus.