Let coordinate of variable point $P(h, k)$.
Given that, coordinate of $A(1,2), B(2,1)$ and
$$
\begin{aligned}
& |P A-P B|=3 \\
& \Rightarrow \sqrt{(h-1)^2+(k-2)^2}-\sqrt{(h-2)^2+(k-1)^2}=3 \\
& \Rightarrow \sqrt{(h-1)^2+(k-2)^2}=3+\sqrt{(h-2)^2+(k-1)^2}
\end{aligned}
$$

Squaring both side
$$
\begin{aligned}
& \Rightarrow(h-1)^2+(k-2)^2 \\
& =9+(h-2)^2+(k-1)^2+6 \sqrt{(h-2)^2+(k-1)^2} \\
& \Rightarrow h^2+1-2 h+k^2+4-4 k \\
& =9+h^2+4-4 h+k^2+1-2 k \\
& +6 \sqrt{(h-2)^2+(k-1)^2} \\
& \Rightarrow 2 h-2 k-9=6 \sqrt{(h-2)^2+(k-1)^2} \\
&
\end{aligned}
$$


Again, squaring both sides
$$
\begin{aligned}
& \Rightarrow(2 h-2 k-9)^2=36\left[(h-2)^2+(k-1)^2\right] \\
& \Rightarrow 4 h^2+4 k^2+81-8 h k+36 k-36 h \\
& =36\left[h^2+4-4 h+k^2+1-2 k\right] \\
& \Rightarrow 4 h^2+4 k^2+81-8 h k+3 k-36 h \\
& =36 h^2+180-144 h+36 k+36-72 k \\
& \Rightarrow 32 h^2+32 k^2+8 h k-108 h-108 k+99=0
\end{aligned}
$$

So, locus of $p(h, k)$ is
$$
32 x^2+32 y^2+8 x y-108 x-108 y+99=0
$$