A point P is outside a circle with radius 8 units. A secant line passing through P and the circle's center O intersects the circle at points A and B, where B is the point on the secant closer to P. A tangent from P touches the circle at T, and PT=15 units. Find the length of the segment PB.
Show answer & explanation
Correct answer: C
Explanation
Given:
Radius of the circle = 8
PT = 15
P, O, A, B are on the same line
By the formula:
Tangent-Sector Theorem
PT² = PB × PA
PT² = 15² = 225
OB = 8 ⇒ AB = 2 × 8 = 16
PA = PB + AB
PT² = PB × (PB + 16)
225 = PB(PB + 16)
225 = PB² + 16PB
PB² + 16PB − 225 = 0
(PB + 25)(PB − 9) = 0
PB = 9