Consider a triangle drawn on the X-Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is
Explanation:
The points satisfying the equations x + y < 41, y > 0, x > 0 lie inside the triangle. Integer solutions of x + y < 41 can be found as follows: If x + y = 40, then
(x, y) could be (1, 39), (2, 38), …, (39, 1) ... (39 solutions)
If x + y = 39, then
(x, y) could be (1, 38), (2, 37), …, (38, 1) ... ( 38 solutions)
If x + y = 38, we get 37 solutions and so on till x + y = 2 ... (1 solution)
∴ Total solutions = 1 + 2 + 3 + ... + 39 = 39 × 40/2 = 780 integer solutions to x + y < 41. The number of points with integer coordinates lying inside the circle = 780
Hence, option (a).
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