Let AB and CD be chords of a circle, that meet at point Q inside the circle. If AQ = 6, BQ = 12, and CD = 30, then find the minimum length of CQ.
Let x be the length of CQ. Then DQ = 30 - x.
By power of points,
x(30 - x) = 72
x^2 - 30x + 72 = 0
(x - 15)^2 - 225 + 72 = 0
(x - 15)^2 = 153
\(x = 15\pm \sqrt{153}\)
Minimum length of CQ = \(15 - \sqrt{153}\)
