# Point to line distance equation

• Oct 2nd 2011, 06:54 AM
BERMES39
Point to line distance equation
The problem says, find the equation that expresses the distance from P(7,7) to a line that crosses the origin and has slope m, as a function of m. Or d(m) = ?

I have found several equations in rectangular and polar coordinates and what the graph should look like, but have been unable to express d as a function of m. I would appreciate some help.
• Oct 2nd 2011, 07:03 AM
HallsofIvy
Re: Point to line distance equation
If a line has slope m, then any line perpendicular to it has slope -1/m. In particular, a line through (7, 7) perpendicular to a line with slope m must be given by y= -(1/m)(x- 7)+ 7.

Of course, a line through the origin with slope m has equation y= mx. So you must solve y= mx= -(1/m)(x- 7)+ 7 for x. Find the y value from y= mx, and then find the distance from that (x, y) to (7, 7).
• Oct 2nd 2011, 08:06 AM
Plato
Re: Point to line distance equation
Quote:

Originally Posted by BERMES39
The problem says, find the equation that expresses the distance from P(7,7) to a line that crosses the origin and has slope m, as a function of m. Or d(m) = ?

Given the point $P(p,q)$ and the line $\ell:Ax+By+C=0$ then the distance from $P\text{ to }\ell$ is $\frac{|Ap+Bq+C|}{\sqrt{A^2+B^2}}$

The line in your question is $mx-y=0$ so the distance is $d(m)=\frac{|7m-7|}{\sqrt{m^2+1}}~.$