A canal runs exactly along the $y$-axis. A pumping station stands at P(-1, 4) and a storage tank at Q(5, 6), the units being kilometres. A control valve must be fitted on the canal at the point that is equally far from the station and the tank.
Where should the valve go?

Any point on the $y$-axis has the form (0, $y$). Equate the squared distances: $(0+1{)}^2+(y-4{)}^2=(0-5{)}^2+(y-6{)}^2$, so $17-8y=61-12y$, giving $4y=44$ and $y=11$. The valve goes at (0, 11), which is $\sqrt{1+49}=\sqrt{50}$ km from the station and $\sqrt{25+25}=\sqrt{50}$ km from the tank. (0, 5) borrows the $y$-coordinate of the midpoint of PQ, but that midpoint is (2, 5), which does not lie on the canal; (11, 0) has the coordinates the wrong way round.