A circle has centre O and radius 9 cm, and the point P satisfies OP = 21 cm. A line through P meets the circle first at A and then at B, with PA = 15 cm. What is the length of PB?

For an external point the power is $O{P}^2-r^2=441-81=360$, and every line through $P$ satisfies PA × PB = 360. So PB = 360/15 = 24 cm. (The chord AB is then 9 cm, well inside the diameter of 18 cm, and the arithmetic is consistent.) The trap 9 cm gives the chord AB = PB − PA rather than the full secant, 30 cm is $OP+r$ and would be right only if the secant ran through the centre, and 6√10 = √360 is the TANGENT length from $P$, not PB.