Chapter 4, Problem 9CR

To determine

To determine the equation of line passing through $\left(2,-3\right)$ and perpendicular to the line $y=-2x-3$ .

Answer to Problem 9CR

The required equation of line passing through $\left(2,-3\right)$ and perpendicular to the line $y=-2x-3$ is $y=\frac{1}{2}x-4$ .

Explanation of Solution

Given information : Given equation of line is $y=-2x-3$ and a point $\left(2,-3\right)$ .

Formula Used:The slope of the perpendicular line is negative reciprocal of slope of original line.

Given equation of line is $y=-2x-3$ .

Here, slope of the line is $-2$ .

Therefore, slope of the perpendicular line will be $-\left(\frac{1}{-2}\right)=\frac{1}{2}$ .

The equation of perpendicular line will be $y=\frac{1}{2}x+c$ .

Since, above perpendicular line passes through the point $\left(2,-3\right)$ .

Therefore, it must satisfy the equation of line.

Then,

  $\begin{array}{c}-3=\frac{1}{2}\times 2+c\\ -3=1+c\\ c=-4\end{array}$

Thus, the equation of required line is $y=\frac{1}{2}x-4$ .