Question

Working with Ellipses:

The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. A planet's distance from the sun is 105,000,000 km at perihelion and 111,000,000 km at aphelion. Find an equation for the planet's orbit. (Place the origin at the center of the orbit with the sun on the x-axis.

Answer 1

center at (0,0)
distance between foci is 111,000,000- 105,000,000 = 6,000,000
therefore c = 3,000,000
a = 3,000,000 + 105,000,000 = 108,000,000
a^2 = b^2 + c^2 --> b^2 = a^2 - c^2
b^2 = (108,000,000)^2 - (3,000,000)^2 = (107,958,325)^2
Equation:
\[\frac{x^{2}}{108,000,000^{2}} + \frac{y^{2}}{107,958,325^{2}} = 1\]

Answer 2

Why isnt it just

x^2/(111)^2 + y^2/(105^2) = 1

Answer 3

Because those distances are from the sun, which is at a focus point.
2a is length of major axis, 2b is length of minor axis
-> x^/a^2 + y^2/b^2 = 1

Answer 4

Q said place origin at centre.

Answer 5

Oh, I see.....

Answer 6

Thank you guys!

Answer 7

It's a lot easier in polar!