Coordinate Geometry.

Let the point O(x,y) be equidistant from the points A (1, 2) , B (3, -4) and C (5, -6)..

Now using distance formula OA = OB = OC

(x-1)² + (y-2)² = (x-3)² + (y+4)²

On squaring and equating, we will get

x + y = 5         ..................(1)

Similarly,

(x-1)² + (y-2)² = (x-5)² + (y+6)²

On squaring and equating, we will get

x - 2y = 7          ...............(2)

Solving the simulatneous linear equations we get the value of x and y as

x = 17/ 3 and y = -2/3

Hence the answer.