jmc

The given equation does not resemble the standard form for a hyperbola, so instead, we appeal to the geometric definition of a hyperbola. Notice that the first term on the left-hand side gives the distance between the points $P=(x,y)$ and $A=(1,-2)$ in the coordinate plane. Similarly, the second term on the left-hand side gives the distance between the points $P$ and $B=(5,-2).$ Therefore, the graph of the given equation consists of all points $P=(x,y)$ such that $$PA-PB=3.$$Thus, by the definition of a hyperbola, the given graph consists of one branch of a hyperbola with foci $A$ and $B.$

The distance between the foci is $AB=4,$ so the distance between each focus and the center is $c=\frac{1}{2}\cdot 4=2.$ Furthermore, if $a$ is the distance between each vertex and the center of the hyperbola, then we know that $2a=3$ (since the general form of a hyperbola is $P{F}_1-P{F}_2=2a$), so $a=\frac{3}{2}.$ Then we have $$b=\sqrt{c^2-a^2}=\frac{\sqrt{7}}{2}.$$The foci $A$ and $B$ lie along a horizontal axis, so the slopes of the asymptotes are $\pm \frac{b}{a}=\pm \frac{\sqrt{7}}{3}.$ The answer is $\boxed{\frac{\sqrt{7}}{3}}.$