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.....

Answer 1

The equation of a hyperbola with center at the origin, transverse axis horizontal, and with distance from the center to a focus (c) of 81,000 km is:

x^2 / a^2 - y^2 / b^2 = 1

where a = 55,000 km is the distance from the center to a vertex.

Since the transverse axis is horizontal, the vertex lies on the x-axis, so b = sqrt(a^2 + c^2) = sqrt(55000^2 + 81000^2) = 98,543 km.

Therefore, the equation of the hyperbola is:

x^2 / (55000)^2 - y^2 / (98543)^2 = 1

Simplified, the equation becomes:

x^2 / 3025000000 - y^2 / 9718811049 = 1

So, the equation that models the path of the satellite (hyperbola) is:

3025000000x^2 - 9718811049y^2 = 3025000000