Chapter 10.5, Problem 4E

To determine

To find: The distance from the given point A to the given line.

Answer to Problem 4E

The distance from the given point A to the given line is $\frac{22\sqrt{17}}{17}\text{unit}$ .

Explanation of Solution

Given information:

The given graph is,

  

The slope of the given line is $-4$ and the y- intercept is $0$ . The slope of the perpendicular line on the given line will be $\frac{1}{4}$ .

The y -intercept of the perpendicular equation on the given line from point A is obtain as,

  $\begin{array}{l}y=mx+c\\ -2=\left(\frac{1}{4}\right)\left(6\right)+c\\ c=-\frac{7}{2}\end{array}$

The equation of the perpendicular equation will be,

  $\begin{array}{l}y=mx+c\\ y=\left(\frac{1}{4}\right)x-\frac{7}{2}\\ 4y=x-14\end{array}$

The intersection point of the equations is at $\left(\frac{14}{17},\frac{56}{17}\right)$ .

Calculate the distance from the given point A to the given line that is point $\left(\frac{14}{17},\frac{56}{17}\right)$ ,

  $\begin{array}{c}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(\frac{14}{17}-6\right)}^2+{\left(-\frac{56}{17}-\left(-2\right)\right)}^2}\\ =\frac{22\sqrt{17}}{17}\text{unit}\end{array}$

Thus, the distance from the given point A to the given line is $\frac{22\sqrt{17}}{17}\text{unit}$ .