Which equation represents a circle in a coordinate plane?

The equation that represents a circle in a coordinate plane is of the form (x-h)² + (y-k)² = r². In this equation, (h, k) represents the center of the circle, while r denotes the radius. This formulation arises from the definition of a circle as the set of all points that are a fixed distance (the radius) away from a central point (the center).

The terms (x-h) and (y-k) represent the horizontal and vertical distances from the center (h, k) to any point (x, y) on the circle. When you square these distances and add them together, you get the squared radius, which geometrically describes all the points equidistant from the center, thereby defining the circle in the coordinate plane.

In contrast, the other options represent different types of mathematical relationships. The linear equation y = mx + b describes a straight line rather than a circle, while A² + B² = C² is the Pythagorean theorem used in right triangles. Lastly, x² + y² = r would not be correct as it suggests a circle centered at the origin with radius sqrt(r), missing the necessary constants for h and k. Thus, option