Find an equation that models the path of a satellite if its path is a hyperbola, a =
55,000 km, and c = 81,000
km. Assume that the center of the hyperbola is the origin and the transverse axis is horizontal.
The general equation of a hyperbola with the center at the origin and the transverse axis horizontal is:
(x^2 / a^2) - (y^2 / b^2) = 1
Given that a = 55,000 km and c = 81,000 km, we can find b using the relationship a^2 + b^2 = c^2:
55,000^2 + b^2 = 81,000^2
3,025,000,000 + b^2 = 6,561,000,000
b^2 = 3,536,000,000
b = √3,536,000,000
b = 59,464.5 km
Therefore, the equation that models the path of the satellite is:
(x^2 / 55,000^2) - (y^2 / 59,464.5^2) = 1