Equation of locus of moving point

Question

How to find the equation of locus of a moving point such that its perpendicular distance from a function $f(x)$ is always $g(x)$?

Answer 1

Consider the plot below:

So, I assumed that $f$ is differentiable on its domain. We know the equation of the line $l$ as follows:

$$l:~~-f'(x_0)x+y+(f'(x_0)x_0-y_0)=A_1x+y+B_1=0$$

Since the distance $d$ between the points $(X,Y)$ and the straight line $l$ is given by $$d=\frac{|A_1X+Y+B_1|}{\sqrt{A_1^2+1}}$$ so it seems that we should consider $$d=g(x)$$