Problem
GEO-B2-M02-P008 A Secant With Known Inner Part
#8
★★★☆☆ Level 3 of 5
From point \(P\), tangent \(PT=12\) and secant \(PAB\) are drawn. It is known that the inner part of the secant is \(AB=20\). Find \(PA\).
If \(PA=x\), then \(PB=x+20\).
Let \(PA=x\). Then \(PB=x+20\). By the tangent-secant formula, \(12^2=x(x+20)\), so \(x^2+20x-144=0\). The positive root is \(x=6\). Hence \(PA=6\).
Here it is important to translate the “inner part” into the whole secant.