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Math Horizons Problem 119 Solution.
by Peter Y. Woo, Biola Univ., 6/1/99
Problem. A circle intersects a rectangular hyperbola in four points. Prove that the sum of the squares of the distances from the points to the center of the hyperbola equals the square of the diameter of the circle.
Solution. (Using theory of equations.) Choose coordinate axes so that
the hyperbola is x y = 1, and the circle is (x - h)2 + (y - k)2 = r2 . . . . .(2)
Let them intersect at the points (xi, yi) for i = 1, 2, 3, 4.
We have to prove that åi=1,...,4 (xi2 + yi2) = 4 r2 . . . . (7)
Substituting 1/x for y into the circle's equation, we get
x4 - 2 h x3 + (h2 + k2 - r2) x2 - 2 k x + 1 = 0 . . . .(3) whose roots are x1, . . . , x4.
By comparing (3) with (x - x1) (x - x2) (x - x3) (x - x4) = 0 . . . . (4),
we see that åxi = 2 h and åxixj = h2 + k2 - r2, so that
åxi2 = (åxi)2 - 2 åxixj = 2 h2 - 2 k2 + 2 r2 . . . . (5)
Similarly, by substituting 1/y for x in (2), we get an equation in y similar to (3), from which we can deduce that
åyi2 = 2 k2 - 2 h2 + 2 r2 . . . . . (6)
Adding (5) and (6) then gives (7), which proves the theorem.
