S' is a region enclosing all such points (x, y) in the X-Y plane, whose distance from the origin is less than or equal to 3√2 units. The number of points, in the region 'S', with integer co-ordinates is

Hi Swaroop,

The number of points required is the number of points with integral coordinates which lie on or inside the circle $$x^2 + y^2 = 18$$

Consider the first quadrant:

$$3\sqrt2$$ = 4.2 approx. The points which lie inside the circle with x = 1 are (1,1), (1,2), (1,3), (1,4)
With x = 2, the points are (2,1), (2,2), (2,3).
(2,4) lies outside because $$2^2 + 4^2 = 20 > 18$$
With x = 3, the points are (3,1), (3,2), (3,3)
With x = 4, the point is (4,1).
(4,2) lies outside.
So, the number of points in the first quadrant is 4+3+3+1 = 11
So, the number of points in all the quadrants = 11*4 = 44
The points on the positive X-axis are (1,0), (2,0), (3,0), (4,0)
The number of points on all the four axes plus origin = 4*4 + 1 = 17
So, the total number of points that lie inside the circle is 44+17 = 61