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.(2 points)
The equation of a hyperbola with center at the origin, transverse axis horizontal, and vertices on the x-axis is:
x^2 / a^2 - y^2 / b^2 = 1
where a = 55,000 km is the distance from the center to each vertex along the x-axis, and c = 81,000 km is the distance from the center to each focus.
First, we can find b using the relationship between a, b, and c in a hyperbola: c^2 = a^2 + b^2
Plugging in the values of a and c, we have:
(81,000)^2 = (55,000)^2 + b^2
6,561,000,000 = 3,025,000,000 + b^2
b^2 = 3,536,000,000
b = √3,536,000,000 ≈ 59,500 km
So, the equation of the hyperbola is:
x^2 / (55,000)^2 - y^2 / (59,500)^2 = 1
x^2 / 3025 x 10^6 - y^2 / 3544 x 10^6 = 1
x^2 / 3025 x 10^6 - y^2 / 3544 x 10^6 = 1