Question

The points of intersection of two equal circles which cut orthogonally are (2,3) and (5,4). Then radius of each circle is?

Answer 1

The line joining the two points is

y-3 = (1/3)(x-2)
Its midpoint is at (7/2,7/2)

The centers of the circles lie on its perpendicular bisector, which is

y-7/2 = -3(x-7/2)

If the radius of the circles is r, the distance from (2,3) to the second line is r/√2

So, just figure that distance, multiply by √2 and you have your radius.

Answer 2

what a lot of bother I went to above.

The distance from (2,3) to (5,4) is √10.

Since the circles are orthogonal at the intersections, the two given points and the centers of the circles form a square, with diagonal √10.

So, the side of the square, which is the radius of the circles, is √5.