It is well known that the equation $y^2=kx$ represents (family of) parabola(s) with axis (of parabola) $y=0$ and tangent at vertex $x=0$.
Suppose instead we are given two mutually perpendicular lines $L_1:\ ax+by+c_1=0$ and $L_2:\ bx-ay+c_2=0$. Then we say that the equation of parabola with $L_1$ as the axis and $L_2$ as the tangent at vertex is $$ \left(\frac{ax+by+c_1}{\sqrt{a^2+b^2}}\right)^2=k\left(\frac{bx-ay+c_2}{\sqrt{a^2+b^2}}\right). $$
One justification for this equation goes like this: the equation $y^2=kx$ can be seen as $$ (\text{distance of $(x, y)$ from the line $y=0$})^2 = k(\text{distance of $(x, y)$ from the line $x=0$}). $$
Then it is said that the second equation is the generalisation of this observation.
But I am not quite convinced on this generalisation. I think that there is some crucial detail missing. Please help me in why the second equation represents the parabola with axis $L_1$ and tangent at vertex $L_2$.