Two sensors are fixed at A(1, 5) and B(4, -2) on a grid. A repeater must stand somewhere on the $x$-axis so that it is the same distance from both sensors.
Give the coordinates of the repeater.

A point on the $x$-axis has the form $(x$, 0). Equate the squared distances: $(x-1{)}^2+(0-5{)}^2=(x-4{)}^2+(0+2{)}^2$, so $x^2-2x+26=x^2-8x+20$, giving $6x=-6$ and $x=-1$. The repeater is at (-1, 0), and both sensors are then $\sqrt{29}$ units away. (0, -1) is the same pair of numbers with $x$ and $y$ swapped, and (2.5, 1.5) is the midpoint of AB, which does not lie on the $x$-axis.