How do I prove it for a general case?

Question

Original question: Let A and B be two points lying on opposite sides of a given line l. Find all point(s) X on l such that |AX − BX| is maximized
My approach: I started by taking a general line l : $y=mx+c$, assumed points A and B to be (a,b) and (c,d) respectively and on either side of the l. To prove that the point X lies on the foot of perpendicular of A and B (I assumed this to be true), I tried to use contradiction and ended up with too many unknowns. Is there a better method to solve this or please tell me if my assumption was wrong.

Dhanvi Sreenivasan · Answer

That doesn't seem to be true. Let us define the hyperbola defined as
$$|d(A,X) -d(B,X)| = c$$
Here we need to find a maximum $c$ given a line $l$ for $X$. My hypothesis is that $l$ would need to be tangent to the curve. Using this, we can find $c$

How do I prove it for a general case?

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