Product Points on a Line

Define coordinates with $A(0,0)$ , $B(p,0)$ and let $C(x,0)$ be a point on the line $AB$ (or its extension). Then $|AC|=|x|$ and $|BC|=|p-x|$ . We want the equation $|x|\cdot |p-x|=q$ to have exactly three real roots. It's easier to square both sides here: $x^2 (p-x)^2=q^2$

For this quartic to have exactly three real roots, one of them must be a double root. The easiest thing to work with is the discriminant, which must be zero for the equation to have a repeated root. This leads to the equation $-16q^4 \left(16q^2-p^4\right)=0$

Now, if $q=0$ , there are exactly two valid points $C$ on the line (ie the points $A$ and $B$ ). So we can divide through to get $16q^2-p^4=0$

Since $p$ is a distance and $q$ is a product of two distances, both are positive; hence $k=\frac{p^2}{q}=\boxed4$ .