Question

In each part that​ follows, you are given an equation of a line and a point. Find the equation of the line through the given point that is perpendicular to the given line.​ (The slope of the perpendicular line is the negative reciprocal of the slope of the given line if the given line is neither vertical nor​ horizontal.)

a) y=9x, P (0, 0)
b) y=3x+4, Q (1, 2)

Answer 1

a)
`y=-(1)/(9)x`

b)
`y-2=-(1)/(3) (x-1)`

Explanation:

We can write the equation of a line in 3 different forms including slope intercept, point-slope, and standard depending on the information we have. We have a point given and a slope from the equation. We will chose point-slope since we have a point and can find the slope.

Point slope:
`y-y_1=m(x-x_1)`

a)
`m\\eq 9` in our new equation because it is perpendicular to it. This means we will need to change it into its negative reciprocal which is
`m=-(1)/(9)`.

We will substitute
`m=-(1)/(9)` and
`x_1=0\\y_1=0`.

`y-0=-(1)/(9) (x-0)`

This simplifies to:

`y=-(1)/(9)x`

b) The equation y=3x+4 follows y=mx+b. For the perpendicular line,
`m\\eq 3`. We will need to change it into its negative reciprocal which is
`m=-(1)/(3)`.

We will substitute
`m=-(1)/(3)` and
`x_1=1\\y_1=2`.

`y-2=-(1)/(3) (x-1)`