Chapter 13, Problem 35RP

To determine

To find: The B point if it lies on the x axis and AB+BC is a minimum.

Answer to Problem 35RP

The point B is $\left(\frac{44}{7},0\right)$ .

Explanation of Solution

Given:

The given points is $A=\left(2,10\right)$ and $C=\left(8,4\right)$ .

Calculation:

Consider that the reflection of C over the x axis is $\left(8,-4\right)$ . Then the line that goes through $\left(2,10\right)$ and $\left(8,-4\right)$ has the slope of,

  $\frac{10-\left(-4\right)}{2-8}=-\frac{7}{3}$

Then, the equation of the line is,

  $y=-\frac{7}{3}x+b$

Solve for the intercept as,

  $\begin{array}{l}0=-\frac{7}{3}x+\left(\frac{44}{3}\right)\\ x=\frac{44}{7}\end{array}$

The point B is $\left(\frac{44}{7},0\right)$ .