What is the equation of a circle with its center at the origin?

The equation of a circle can be defined based on its center and radius. For a circle centered at the origin (0,0), the equation takes the form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius. In this case, since the center is at the origin, h = 0 and k = 0. Therefore, the equation simplifies to x² + y² = r², indicating that the radius of the circle is r.

This correct choice articulates that the sum of the squares of the coordinates of any point (x, y) on the circle is equal to the square of the radius (r) squared, reinforcing the geometric definition of a circle. This implies that any point on the circle maintains a constant distance (the radius) from the center.

The other options do not correctly describe a circle centered at the origin. The second choice utilizes a variable 'r' instead of 'R' but still implies a similar concept. The third choice incorrectly introduces a difference of squares which does not represent a circle but rather hyperbolic equations. The fourth choice presents an equation that does not maintain the necessary structure