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

Since the transverse axis is horizontal, the equation of the hyperbola can be written as:

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

where a = 55000 km and c = 81000 km. We can find b using the relationship between a, b, and c in a hyperbola:

c^2 = a^2 + b^2
b^2 = c^2 - a^2
b^2 = (81000)^2 - (55000)^2
b^2 = 6561000000 - 3025000000
b^2 = 3536000000
b = sqrt(3536000000)
b = 59429.82

Therefore, the equation of the hyperbola that models the path of the satellite is:

x^2 / (55000)^2 - y^2 / (59429.82)^2 = 1