How Many Integer Coordinate Points Lie on the Circumference of Circle x^2 y^2 25?

To find the integer coordinate points that lie on the circumference of the circle defined by the equation x^2 y^2 25, we need to determine integer pairs (x, y) such that this equation holds true.

The equation represents a circle with radius 5 centered at the origin (0, 0). We can find the integer solutions by considering the possible integer values for x and calculating the corresponding y values.

Step-by-Step Analysis:

Possible Values for x:

Since x^2 must be less than or equal to 25, x can take values from -5 to 5. That is, x -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.

Calculating Corresponding y Values:

If x 0:

Y2 25 implies y ±5 → (0, 5), (0, -5)

If x ±1:

Y2 25 - 12 24 → no integer solutions

If x ±2:

Y2 25 - 22 21 → no integer solutions

If x ±3:

Y2 25 - 32 16 implies y ±4 → (3, 4), (3, -4), (-3, 4), (-3, -4)

If x ±4:

Y2 25 - 42 9 implies y ±3 → (4, 3), (4, -3), (-4, 3), (-4, -3)

If x ±5:

Y2 25 - 52 0 implies y 0 → (5, 0), (-5, 0)

Summary of Integer Solutions:

From x 0: (0, 5), (0, -5) From x ±3:(3, 4), (3, -4), (-3, 4), (-3, -4) From x ±4: (4, 3), (4, -3), (-4, 3), (-4, -3) From x ±5: (5, 0), (-5, 0)

Total Count:

Counting all unique integer coordinate points on the circle, we find there are 12 points in total.

Conclusion:

The 12 points with integer coordinates that lie on the circumference of the circle x^2 y^2 25 are: (0, 5), (0, -5), (3, 4), (3, -4), (-3, 4), (-3, -4), (4, 3), (4, -3), (-4, 3), (-4, -3), (5, 0), (-5, 0).

Understanding these integer coordinate points is crucial for various applications in mathematics, particularly in number theory and geometry. For more detailed analysis and further exploration, consider diving into the broader field of integer lattice points on circles and their applications.