Chapter 13.5, Problem 23WE

To determine

To find: the coordinates of $F$

Answer to Problem 23WE

The coordinates of $F$ is $\left(\frac{5x_1}{8}+\frac{3x_2}{8},\frac{5y_1}{8}+\frac{3y_2}{8}\right)$

Explanation of Solution

Given information: $F$ is on $\overline{PQ}$ and $PF=\frac{3}{8}PQ$ . If $P\left(x_1,y_1\right)\&Q\left(x_2,y_2\right)$ where $x_1<x_2$

Calculation:

  $\begin{array}{l}P\left(x_1,y_1\right)\&Q\left(x_2,y_2\right)\\ PF=\frac{3}{8}PQ\end{array}$

Since, $PF$ is $\frac{3}{8}$ times $PQ$ which makes $FQ$ $\frac{5}{8}$ times $PQ$

So ratio,

  $\begin{array}{l}PF:FQ\\ 3:5\end{array}$

  $\begin{array}{l}F\left(x_3,y_3\right)=\left(\frac{5x_1+3x_2}{5+3},\frac{5y_1+3y_2}{5+3}\right)\\ F\left(x_3,y_3\right)=\left(\frac{5x_1}{8}+\frac{3x_2}{8},\frac{5y_1}{8}+\frac{3y_2}{8}\right)\end{array}$