Chapter 13.7, Problem 33WE

To determine

To find : the value of $k$

Answer to Problem 33WE

The values of $k$ are $\frac{2}{3},-\frac{2}{3}$

Explanation of Solution

Given information :

The line $y=9kx-1$ and $kx+4y=12$ are perpendicular

Formula used :

  $y=mx+b$

Calculation :

The line $y=9kx-1$

Therefore, the slope of line $m=9k$

The line is

  $\begin{array}{l}kx+4y=12\\ 4y=-kx+12\\ y=-\frac{k}{4}x+3\end{array}$

Therefore, the slope of another line $m=-\frac{k}{4}$

The product of the slopes must be equal to $-1$ , because the both lines are perpendicular.

  $\begin{array}{l}9k\cdot \left(-\frac{k}{4}\right)=-1\\ -\frac{9}{4}{k}^2=-1\\ \frac{9}{4}{k}^2=1\\ k^2=\frac{4}{9}\\ k=\sqrt{\frac{4}{9}}\\ k=\pm \frac{2}{3}\\ k=\frac{2}{3},-\frac{2}{3}\end{array}$

Hence, the values of $k$ are $\frac{2}{3},-\frac{2}{3}$ .