jmc

Question

If A = (3,4) and C = (7,10) are opposite vertices of a rectangle ABCD, then vertices A, B, C, and D must lie on the circle x^2 + y^2 - px - qy + s = 0.Compute the ordered triple of real numbers (p,q,s).

Accepted Answer

Since ABC = ADC = 90^, AC must be a diameter of this circle. FIGURE The center of the circle is then (5,7) (the mid-point of A and C), and its radius is (5 - 3)^2 + (7 - 4)^2 = 13, so its equation is (x - 5)^2 + (y - 7)^2 = 13.This simplifies to x^2 + y^2 - 10x - 14y + 61 = 0, so (p,q,s) = (10,14,61).