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Question

The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is midway along its orbit, as shown, how far is it from the sun, in AU?

Accepted Answer

Let A be the perigee, let B be the apogee, let F be the focus where the sun is, let O be the center of the ellipse, and let M be the current position of Xavier. FIGURE Then AB is a major axis of the ellipse, and AB = 2 + 12 = 14. Since M is the midway point, MF = AO = 142 = 7.