Q:

How to find equation of line rotated $\theta$ degree counterclockwise about $P$?

Let $P$ is a point on line $ax+by+c=0$. then 

How to find equation of line rotated $\theta$ degree counterclockwise about $P$?

A:

As parametric equations of the line 
$$
r=\{(x,y)\in \mathbb{R}^2: a\cdot x+b\cdot y+c=0 \}
$$ 
that passes through the point $P_0=(x_0,y_0)$ point are given by
$$
\left\lgroup
\begin{array}{c}
x(t)
\\
y(t)
\end{array}
\right\rgroup
=
\left\lgroup
\begin{array}{c}
x_0+at
\\
y_0+bt
\end{array}
\right\rgroup
$$
Let 
$$
\left\lgroup
\begin{array}{c}
x_\theta(t)
\\
y_\theta(t)
\end{array}
\right\rgroup
$$
parametric coordenates of a line rotated $\theta$ degree counterclockwise about $P_0=(x_0,y_0)$.Then
$$
\left\lgroup
\begin{array}{c}
x_\theta(t)
\\
y_\theta(t)
\end{array}
\right\rgroup
=
\left\lgroup
\begin{array}{cc}
\cos \theta & \sin\theta
\\
-\sin\theta & \cos\theta
\end{array}
\right\rgroup
\left\lgroup
\begin{array}{c}
x(t)-x_0
\\
y(t)-y_0
\end{array}
\right\rgroup
+
\left\lgroup
\begin{array}{c}
x_0
\\
y_0
\end{array}
\right\rgroup
$$

