Question

Which of the following equations contains the point (8, 5) and is perpendicular to the line y = 2x − 3?

Answer 1

`y = \ -\displaystyle(1)/(2)x \ + \ 9`

Explanation:

Given that the reference line is y = 2x - 3, with a slope of
`m_(reference) \ = \ 2`.

We know that two non-vertical lines are perpendicular if the slope of one line is the negative reciprocal of the slope of the other. In other words, both slopes can be multiplied together to yield -1. Let
`m_(1)` be the slope of one line and
`m_(2)` be the slope of its corresponding perpendicular line,

`m_(1) \ * \ m_(2) \ = \ -1 \ \ \ \ \ \ \ \ \ \ \ \mathrm{or} \ \ \ \ \ \ \ \ \ \ \ m_(1) \ = \ \displaystyle(-1)/(m_(2))`.

Thus,

`m_(reference) \ * \ m_(perpendicular) \ = \ -1 \\ \\ \-\hspace{2.26cm} m_(perpendicular) \ = \ \displaystyle(-1)/(m_(reference)) \\ \\ \-\hspace{2.26cm} m_(perpendicular) \ = \ \displaystyle(-1)/(2) \\ \\ \-\hspace{2.26cm} m_(perpendicular) \ = \ -\displaystyle(1)/(2)`

Therefore, using the point-slope form for the equation of a line passing through the point
`(x_(1), \ y_(1))` is
`y \ - \ y_(1) \ = \ m(x \ - \ x_(1))`. Given that the perpendicular line passes through the point
`(8,\ 5)`, the equation of the perpendicular line is

`y \ - \ 5 \ = \ -\displaystyle(1)/(2)(x \ - \ 8) \\ \\ y \ - \ 5 \ = \ -\displaystyle(1)/(2)x \ + \ 4 \\ \\ \-\hspace{0.85cm} y \ = \ -\displaystyle(1)/(2)x \ + \ 9`