Thread: Find the equation of a parallel line

1. 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?

2. The slope of the line $\displaystyle mx+ny+c=0$ is $\displaystyle -\frac{m}{n}$

The equation of the line is

$\displaystyle y-3=-\frac{m}{n}(x-2)\Rightarrow m(x-2)+n(y-3)=0$

3. Originally Posted by fcabanski
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

4. Why are the x and y values subtracted from x and y in red dog's answer?

5. Originally Posted by fcabanski
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

$\displaystyle y-y_0=m(x-x_0)$

6. Originally Posted by fcabanski
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 $\displaystyle D=-(2M+3N)$
$\displaystyle Mx+Ny-(2M+3N)=0$
$\displaystyle M(x-2)+N(y-3)=0$

7. Thanks everyone!