Circles and Chords

Geometry Level 5

A tangent is drawn on a circle at the point of contact $X$ . On this tangent, a variable point $P$ is chosen. A line is then drawn through $P$ such that it intersects the circle at two points $A$ and $B$ (not necessarily distinct) where $B$ is further away from $P$ . Another circle centred at $P$ with radius $PB$ is drawn, and the line $BX$ (extended if necessary) intersects it at the points $B$ and $C$ .

It is given that $BX = 8$ and that $XC = 12$ . Calculate the mean value of $(AB)(PB)$ as $P$ moves on the tangent.

The answer is 96.

1 solution

Michael Ng
Nov 25, 2014

We prove that $(AB)(PB) = (BX)(XC) = 96$ .

By the intersecting chords theorem on the second circle: $(BX)(XC) = (PB + PX)(PB - PX) = PB^2 - PX^2$ (This result may also be known as the power of a point theorem.)

Now by the tangent-secant theorem on the first circle, $PX^2 = (PA)(PB)$ .

Substituting in gives: $(BX)(XC) = PB^2 - (PA)(PB) = PB(PB - PA) = (AB)(PB)$ as required. Hence the answer is $\boxed{96}$ .

Circles and Chords

The answer is 96.

1 solution

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