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Math Help - Point to line distance equation

  1. #1
    Junior Member BERMES39's Avatar
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    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.
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  2. #2
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    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).
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    Re: Point to line distance equation

    Quote Originally Posted by BERMES39 View Post
    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}}~.
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