Question

Find the equation for the line that passes through the point (5,-3), and that is perpendicular to the line with the equation y-1=1/4(x-2)

Answer 1

Answer:

y+3=-4(x-5)

Explanation:

Part A

Given the line:

$y-1=\frac{1}{4}(x-2)$

We want to find the equation of a perpendicular line that passes through the point (5,-3),

First, determine the slope of the perpendicular line.

Comparing the given line with the slope-point form:

$\begin{gathered} y-y_1=m(x-x_1) \\ \implies\text{Slope},m=\frac{1}{4} \end{gathered}$

By definition, two lines are perpendicular if the product of their slopes is -1.

Let the slope of the perpendicular line = n

$\begin{gathered} \implies\frac{1}{4}n=-1 \\ n=-4 \end{gathered}$

Thus, using a slope of -4 and a point (5,-3), we find the equation of the line.

$\begin{gathered} y-y_1=m(x-x_1) \\ y-(-3)=-4(x-5) \\ y+3=-4(x-5) \end{gathered}$

The equation of the perpendicular line in the slope-point form is:

$y+3=-4(x-5)$

Part B

In order to graph the line, first, find two points on the line.

When x=0

$\begin{gathered} y+3=-4(0-5) \\ y+3=20 \\ y=20-3=17 \\ \implies(0,17) \end{gathered}$

When y=1

$\begin{gathered} 1+3=-4(x-5) \\ 4=-4(x-5) \\ \frac{4}{-4}=x-5 \\ -1=x-5 \\ x=5-1=4 \\ \implies(4,1) \end{gathered}$

Join the points (0,17) and (4,1) as shown in the graph below: