In the circular park shown, $AB$ is a diameter of length $13$ m and the pole is placed at point $P$ on the boundary such that the difference of its distances from gates $A$ and $B$ is $7$ m. At what distances from $A$ and $B$ should the pole be erected? 

Since $AB$ is a diameter, $\angle APB=90^\circ$, so triangle $APB$ is right-angled at $P$. Let $BP=x$ m, then because the difference is $7$ m, $AP=x+7$. By Pythagoras, $(x+7)^2+x^2=13^2$, which simplifies to $x^2+7x-60=0$. Solving gives $x=5$ or $x=-12$; only the positive value is valid, so $BP=5$ m and $AP=12$ m. Hence the pole should be $12$ m from $A$ and $5$ m from $B$.