Find the equation of a line which is parallel to the linemx +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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- Jul 1st 2009, 01:05 PMfcabanskiFind the equation of a parallel line
Find the equation of a line which is parallel to the line

*m*x +*n*y +*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?

- Jul 1st 2009, 01:09 PMred_dog
The slope of the line $\displaystyle mx+ny+c=0$ is $\displaystyle -\frac{m}{n}$

The equation of the line is

$\displaystyle y-3=-\frac{m}{n}(x-2)\Rightarrow m(x-2)+n(y-3)=0$ - Jul 1st 2009, 01:13 PMCaptainBlack
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 - Jul 1st 2009, 01:56 PMfcabanski
Why are the x and y values subtracted from x and y in red dog's answer?

- Jul 1st 2009, 02:00 PMred_dog
- Jul 1st 2009, 02:06 PMPlato
**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 $\displaystyle D=-(2M+3N)$

$\displaystyle Mx+Ny-(2M+3N)=0$

$\displaystyle M(x-2)+N(y-3)=0$ - Jul 1st 2009, 02:11 PMfcabanski
Thanks everyone!