Question

Consider the line ="absmiddle" class="latex-formula">
Find the equation of the line that is perpendicular to this line and passes through the point $(5 , 2)$

Find the equation of the line that is parallel to this line and passes through the point $( 5 , 2)$

Equation of perpendicular line:

Equation of parallel line:

Answer 1

Firstly we put the equation of the line into $y=mx+c$ form so we can find the gradient.

$-5x-4y=7 \Rightarrow -5x-7=4y \Rightarrow \frac{-5}{4}x-\frac{7}{4}=y$

So we see the gradient of the line is -5/4. The gradient of a line perpendicular to a line with gradient m is $-\frac{1}{m}$ so we see the gradient of a line perpendicular to your line is $-\frac{1}{-\frac{5}{4}} = \frac{4}{5}$.

So we know it's a line with gradient 4/5 passing through (5,2) hence

$y-2=\frac{4}{5}(x-5) \Rightarrow y-2=\frac{4}{5}x-4 \Rightarrow y=\frac{4}{5}x-2$.

For the parallel line, you use the same gradient as the original which is -5/4 and using the fact that it passes through (5,2) repeat the process from above to find the line. I'll leave this one up to you!