Mathematics

The key insight is that for any point on an ellipse, the sum of distances to the two foci is constant and equals $2a$. Start by using the origin—which lies on the ellipse—to find this sum, then calculate the distance between the foci to get $2c$. Once you have both, eccentricity $e = c/a$ follows directly from their ratio.

The key insight is that for any point on an ellipse, the sum of distances to the two foci is constant and equals $2a$. Start by using the origin—which lies on the ellipse—to find this sum, then calculate the distance between the foci to get $2c$. Once you have both, eccentricity $e = c/a$ follows directly from their ratio.