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Question

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

To find the equation of a hyperbola when the center is at the origin and the transverse axis is horizontal, we use the formula:

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

where a = 55,000 km and c = 81,000 km. We can find b using the relationship b^2 = c^2 - a^2.

b^2 = 81000^2 - 55000^2
b^2 = 6561000000 - 3025000000
b^2 = 3536000000
b = √3536000000
b ≈ 59464.82 km

Now we can write the equation of the hyperbola:

x^2 / 55000^2 - y^2 / 59464.82^2 = 1

Simplifying this equation, we get:

x^2 / 3025000000 - y^2 / 3536000000 = 1

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

x^2 / 3025000000 - y^2 / 3536000000 = 1