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Thread: Find the equation of a parallel line

  1. #1
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    Find the equation of a parallel line

    Find the equation of a line which is parallel to the line mx + ny + c = 0 and passing through the point (2,3).


    The given answer is: m(x-2)+n(y-3) = 0, but how do I get there?
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    MHF Contributor red_dog's Avatar
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    The slope of the line $\displaystyle mx+ny+c=0$ is $\displaystyle -\frac{m}{n}$

    The equation of the line is

    $\displaystyle y-3=-\frac{m}{n}(x-2)\Rightarrow m(x-2)+n(y-3)=0$
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by fcabanski View Post
    Find the equation of a line which is parallel to the line mx + ny + c = 0 and passing through the point (2,3).


    The given answer is: m(x-2)+n(y-3) = 0, but how do I get there?
    The lines $\displaystyle mx+ny+c=0$ for fixed $\displaystyle n$ and $\displaystyle m$ form a series of parallels as $\displaystyle c$ varies.

    So the line you seek is found by finding the value of $\displaystyle c$ so that $\displaystyle mx+ny+c=0$ goes through $\displaystyle (2,3)$. That is:

    $\displaystyle 2m+3n+c=0$

    or:

    $\displaystyle c=-(2m+3n)$

    CB
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    Why are the x and y values subtracted from x and y in red dog's answer?
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    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by fcabanski View Post
    Why are the x and y values subtracted from x and y in red dog's answer?
    The equation of a line passing through the point $\displaystyle M_0(x_0,y_0)$ and having the slope $\displaystyle m$ is

    $\displaystyle y-y_0=m(x-x_0)$
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  6. #6
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    Quote Originally Posted by fcabanski View Post
    Find the equation of a line which is parallel to the line mx + ny + c = 0 and passing through the point (2,3).
    Any line parallel to $\displaystyle Mx+Ny+C=0$ has the same form $\displaystyle Mx+Ny+D=0$.
    If it contains the point $\displaystyle (2,3)$ this must be true:
    $\displaystyle 2M+3N+D=0$ or $\displaystyle D=-(2M+3N)$
    $\displaystyle Mx+Ny-(2M+3N)=0$
    $\displaystyle M(x-2)+N(y-3)=0$
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  7. #7
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    Thanks everyone!
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