   SEARCH HOME Math Central Quandaries & Queries  Question from shabnam, a student: the line presented by y= 3x-2 and a line perpendicular to it intersect at R(1,1). Determine the equation of the perpendicular line Hi Shabnam.

There are two parts to your question: (a) determining the slope of the new line and (b) determining the equation of the new line with the point given and the slope calculated in part (a).

The slope of a line that is perpendicular to another is the negative reciprocal of the other. So if line 1 has a slope of 2, then the perpendicular line has a slope of -1/2. If the first line has a slope of -3/7, then the perpendicular line has a slope of 7/3.

Can you read the slope from the equation of the first line? y = 3x - 2 is in the y = mx + b form, and m is the slope, so the slope of the given line is 3.

What do you think, then, is the perpendicular slope to this line? That's part (a).

Now you can find the equation of the new line. You know its slope and you have the point (1, 1). So you write a new line equation y = mx + b and you substitute in the slope you got in (a) and the point (1, 1) which means x=1, y=1. That gives you a simple equation with just numbers and b. So you can solve for b. Then you can write it all out again in y = mx + b form with the b value you found and the m value you found.

Here's an example:
Problem: find the equation of the line perpendicular to the line y = -4x + 1 which intersects it as (2, -7).

Solution:
The given line has slope m = -4.
Therefore the perpendicular line's slope is (1/4), because that is the negative reciprocal of -4.
y = mx + b .... I substitute (1/4) for m, and the point (2, -7) means substitute 2 for x and -7 for y. Then solve for b:
-7 = (1/4)(2) + b
-7 = 1/2 + b
b = -15/2
Finally, I can write the new line in y = mx + b substituting the m and the b, but leaving the x and the y alone:
y = (1/4)x - 15/2
Finished.

Now you try it with your equation y = 3x - 2 and point (1,1).

Cheers,
Stephen La Rocque     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.