Did you mean to say 'How many integral values x and y exist such that (x,y) lies on the circle'?

yup, ans is 12

Radius of the circle is 5.
So any point on the circle with coordinates (x, y) must satisfy x² + y² = 5² = 25

Now, 25 can be expressed as the sum of two integers only in following two ways
0² + (±5)² = 5² and (±3)² + (±4)² = 5²

So, number of possible values of (x, y) = {(0, ±5) : 2 values} + {(±5, 0) : 2 values} + {(±3, ±4) : 4 values} + {(±4, ±3) : 4 values} = 12