Chapter 9.7, Problem 19WE

To determine

The value of PB from the given figure

Answer to Problem 19WE

The value of PB is 4

Explanation of Solution

Given information:

It is given PD=5, CD=7, AB=11 and a figure,

  

Let PB = x

When a secant segment and a tangent segments are drawn to a circle from an external point then the product of the secant segment and its external segment is equal to the square of the tangent segment.

Therefore, from the given figure

  $P{T}^2=PD\times PC$

Also

  $P{T}^2=PB\times PA$

Comparing and solving both the equations

  $\begin{array}{c}PD\times PC=PB\times PA\\ 5\times \left(5+7\right)=x\times \left(x+11\right)\\ 5\times 12=x^2+11x\\ x^2+11x-60=0\\ x^2-4x+15x-60=0\\ x\left(x-4\right)+15\left(x-4\right)=0\\ \left(x-4\right)\left(x+15\right)=0\end{array}$

This gives x = 4 or x = -15

Since length is never negative. So we reject x = -15

Therefore the value of PB = 4