How can we convert $ax + by + c = 0$ to $x\cos α + y\sin α = p$? Here $α$ is the angle of perpendicular $p$ w.r.t $x$-axis.
I compared both equations and got $ax = x \cos α$, $by = y \sin α$ and $c = -p$, and got $a = \cos α$ and $b = \sin α$. I do not know what to do afterward.
Hint:
A cartesian equation for a straight line is defined up to a non-zero factor. This means that the equations $ax+by+c=0$ and $x\cos\alpha+y\sin\alpha -p=0$ represent the same line if and only if $$\frac{\cos\alpha}a=\frac{\sin\alpha}b=-\frac pc.$$ The first equality implies $\;\dfrac{\cos^2\alpha}{a^2}=\dfrac{\sin^2\alpha}{b^2}=\dfrac{\cos^2\alpha+\sin^2\alpha}{a^2+b^2}=\cdots$.
