Two villages lie at P(-7, 5) and Q(5, -1) on a map where one unit is 1 km…

Call the moving point $P(x, y)$, write the rule as an equation, and square to clear any root. $PA = PB$ gives the perpendicular bisector of $AB$: $2(x_2 - x_1)x + 2(y_2 - y_1)y = x_2^2 + y_2^2 - x_1^2 - y_1^2$. The distance of $P$ from $ax + by + c = 0$ is $\dfrac{|ax + by + c|}{\sqrt{a^2 + b^2}}$. If $P$ is the midpoint of $A(a, 0)$ and $B(0, b)$, then $a = 2x$ and $b = 2y$. $PA \perp PB$ means the slopes multiply to $-1$.