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 and transverse axis horizontal is:

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

where a and b are the lengths of the semi-major and semi-minor axes, respectively, and c is the distance from the origin to each focus, given by the equation:

c = sqrt(a^2 + b^2)

In this case, a = 55,000 km and c = 81,000 km, so we can solve for b:

c^2 = a^2 + b^2
b^2 = c^2 - a^2
b^2 = (81,000 km)^2 - (55,000 km)^2
b ≈ 52,225 km

Now we can substitute in the values of a, b, and the center of the hyperbola (0,0) to get the equation:

x^2/(55,000 km)^2 - y^2/(52,225 km)^2 = 1

So the path of the satellite can be modeled by this hyperbolic equation.