Question

12. Write the equation of the line that is perpendicular to the line x - 4y = 20 and passes through the point (2,-5).

Answer 1

Two perpendicular lines have reciprocal and opposite slopes.

First we have to write the given line in the slope-intercept form:

`y=mx+b`

Where m is the slope and b is the y-intercept.

We have this equation:

`x-4y=20`

To write it in the slope-intercept form we have to clear y:

`\begin{gathered} x-20=4y \\ \downarrow \\ y=(1)/(4)x-5 \end{gathered}`

The slope is 1/4 and the y-intercept is -5.

The slope of the perpendicular line will be the opposite and reciprocal of 1/4, that's -4.

For now we have the perpendicular line's equation:

`y_p=-4x+b`

There are a lot of lines that are perpendicular to the given line, but only one that passes through (2, -5). We use this point to find the y-intercept by replacing x = 2 and y = -5 into the expression above and solving for b:

`\begin{gathered} -5=-4\cdot2+b \\ -5=-8+b \\ -5+8=b \\ b=3 \end{gathered}`

The y-intercept of the perpendicular line is 3.

The equation of a line perpendicular to the given line that passes through the point (2,-5) is

`y_p=-4x+3`