Conics Word Problem

Question

The distance of Negoiu from its star, XO-1, ranges from 566 million mi to 906 million mi. Find the equation of its elliptical orbit where the star is one of the foci. Assume that the center is the origin and its major axis is the y axis.

Answer 1

If c is the maximum y value of the ellipse, b is the maximum x value of the ellipse, and a is the y value of the larger focal point:

c - a = 566

906 = c + 2a

c = 566 + a

906 = 566 + 3a

340 = 3a

340/3 = a

c = 566 + 340/3 = 2048/3

b^2 = c^2 - a^2

b ≈ 689

(x/c)^2 + (y/b)^2 = 1

Answer 2

y²/906² + x²/566² = 10¹² (x and y in milions miles)