Move point on line defined by general form of equation

I have line defined by general form of equation: $$Ax + By + C = 0.$$

I know coordinates of point $P$ [$P_x$, $P_y$]. The point $P$ is on the line.

How can I find coordinates of points $P'$ and $P''$ in specific distance $d$ from the point $P$?

$|P'P| = d = |PP''|$

------------P'------P------P''------------------ <- line Ax + By + C = 0
<------>|<------>
d   |   d

• you should solve a system of equations obtained by noticing that these two points are just the intersection of a given circle and a given straight line (notice that the equations are not linear) – aflous May 7 '14 at 9:46

Let $\left(u,v\right)$ be the coördinates of $P$.
Then any point $P'$ on the line can be written as $\left(u,v\right)+\lambda\left(B,-A\right)$ for some $\lambda\in\mathbb{R}$.
Here $\overline{PP'}=d$ means that $\left\Vert \lambda\left(B,-A\right)\right\Vert =d$ or equivalently $$|\lambda|=d\times\left\Vert \left(B,-A\right)\right\Vert ^{-1}=\frac{d}{\sqrt{A^{2}+B^{2}}}$$
This enables you to find the coördinates of $P'$.