jmc

Question

Let P be a point outside of circle O. A segment is drawn from P such that it is tangent to circle O at point T. Meanwhile, a secant from P intersects O at A and B, such that PA < PB. If PA = 3 and PT = AB - PA, then what is PB?

Accepted Answer

First of all, we see that PB = PA + AB = 3 + AB. By Power of a Point, we know that (PA)(PB) = (PT)^2, so we have 3(PB) = (AB - 3)^2. FIGURE Let us define x such that x = PB = 3 + AB, then AB = x - 3. Substituting, we now have 3x = (x - 6)^2. Then, we see that 3x = x^2 - 12x + 36, so x^2 - 15x + 36 = 0. Factoring, we have (x - 3)(x - 12) = 0 so x = 3 or x = 12, but we are given that PA < PB, so x > 3. That means our only answer for x, hence PB, is 12.