Question

How do you determine the equation of a line passing through (2, -3) that is perpendicular to the line 4x-y=22?

Answer 1

Equation of a line passing through `(2, -3)` that is perpendicular to the line `4x-y=22` is `x+4y+10=0`

Explanation:

Equation of a line that is perpendicular to a line `ax+by+c=0`, will be of type

`bx-ay+k=0`

Note that coefficients of `x` and `y` have changed and sign before `y` has been changed, keeping sign before `x` to be same.

Hence, equation of a line that is perpendicular to a line `4x-y=22`, will be of type

`x+4y+k=0`

As it passes through `(2,-3)`, we have `2+4xx(-3)+k=0` or `2-12+k=0` i.e. `k=12-2=10`.

Hence, equation of a line passing through `(2, -3)` that is perpendicular to the line `4x-y=22` is `x+4y+10=0`.
graph{(4x-y-22)(x+4y+10)=0 [-7.89, 9.89, -5.564, 3.325]}
Note: Equation of a line that is parallel to a line `ax+by+c=0`, will be of type `ax+by+k=0`, i.e. no change in coefficients or sign. Only constant term changes. Also observe that slopes of parallel llines are equal but product of slopes of perpendicular lines is `-1`.