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Math Horizons Problem 115 Solution.
by Peter Y. Woo, Biola Univ., 6/2/99
Problem. (Proposed by Peiyi Zhao, St. Cloud State Univ.) Let Pi, i = 1..n be n variable points on the ellipse E: x2/a2 + y2/b2 = 1, centered O, such that each line OPi is at constant angle 2p/n with the next line OPi+1. Prove that å(1/OPi2) is constant as the points Pi move on the ellipse.
Solution. Let each Pi be at (xi, yi) and let xi = ricos qi, yi = risin qi.
Then Pi lies on E iff ri2 ( cos2qi/a2 + sin2qi/b2) = 1.
\We need to prove å(1 / ri2) = constant, which is equivalent to proving
(å cos2qi) /a2 + (å sin2qi) /b2 = constant.
Now let qi = a + ib, where b = 2p/n.
Then åcos2qi = åi=1..n(1 + cos 2qi)/2
= n/2 + (å cos 2(a + ib) sin b) / 2 sin b
= n/2 + (-sin (2a + b) + sin(2a+ (2 n + 1)b)) / 4 sin b
= n/2 + (-sin (2a + b) + sin(2a+ b)) / 4 sin b because 2 n b = 4p,
= n/2.
Similarly, åsin2qi = n/2 also. Hence
(å cos2qi) /a2 + (å sin2qi) /b2 = n = constant.
