Anlaytic Geometry Help! (Hyperbola)?

Question

A comet follows a hyperbolic orbit about the sun, reaching its closest point to the sun at a vertex of 43 million miles. When the line joining the sun and the comet is perpendicular to the transverse axis of the hyperbola, the comet is 137 million miles from the sun. Give an equation for the comet's orbit if the axes are placed with the x-axis on the transverse axis and the origin at the center. Where is the sun (point)?

Answer 1

The equation which proves most useful in this example is:
Polar coordinates centered at focus,(C, 0) [<===sun]
r = a(e^2 - 1)/(1 - e cosθ)
Using this formula, the given point, θ = 180, r = 4.3x10^7 yields an expression for e - 1 in terms of a:
r = a(e^2 - 1)/(1 - e cosθ)
4.3x10^7 = a(e + 1)(e - 1)/(1 + e) = a(e - 1)
e - 1 = 4.3x10^7/a [<=== (1)]
The other given point,θ = 90, r = 1.37x10^8, yields:
1.37x10^8 = a(e^2 - 1) [<=== (2)]
Substituting (1) into (2):
1.37x10^8 = a(e + 1)(4.3x10^7)/a
e + 1 = 1.37x10^8/4.3x10^7 = 3.186
e = 2.186
Using equation (1):
a = 4.3x10^7/(2.186 - 1) = 3.6256x10^7
Then use the equation relating a, b, and excentricity e:
b = a √(e^2 - 1) = 3.6256x10^7√(2.186^2 - 1) = 7.0477x10^7
Now just plug a and b into the standard equation of a hyperbola:
x^2/(3.6256x10^7)^2 - y^2/(7.0477x10^7)^2 = 1