How do you find the equation that models the path of a satellite moving in a hyperbolic trajectory given the values of a and c?

Finding the Equation of a Hyperbola

To find the equation of a hyperbola that models the path of a satellite, we first need to understand the parameters given. In this case, we have:

a = 55000 km (the distance from the center to the vertex along the transverse axis)
c = 81000 km (the distance from the center to the foci)

The relationship between a, b, and c in a hyperbola is given by the equation:

c² = a² + b²

From the values of a and c, we can find b (the distance from the center to the co-vertices) using the rearranged formula:

b = sqrt(c² – a²)

Let’s calculate b:

B = sqrt(81000² – 55000²)
= sqrt(6561000000 – 3025000000)

= sqrt(3536000000) ≈ 59594.45 km

Now that we have values for a and b, we can put them into the standard equation for a hyperbola:

The standard form of a hyperbola centered at the origin is:

(x² / a²) - (y² / b²) = 1

Since we want the hyperbola centered at the origin (0, 0), our hyperbola can be defined as:

(x² / (55000²)) – (y² / (59594.45²)) = 1

This is the equation that models the path of the satellite moving in a hyperbolic trajectory.