Which of the following represents the standard form of the equation of a circle centered at (h, k) with radius r?

The standard form of the equation of a circle centered at the point (h, k) with radius r is given by the formula ((x - h)^2 + (y - k)^2 = r^2). This equation is derived from the definition of a circle in a Cartesian coordinate system, where the set of all points (x, y) that are a distance r away from the center (h, k) satisfy this relationship.

In this equation, the term ((x - h)^2) represents the horizontal distance of a point on the circle from the center, while the term ((y - k)^2) represents the vertical distance. When you sum these two squared distances, you get r squared, which is the square of the radius of the circle.

Both the correct formula and the concept reinforce how a circle encompasses all constant distances from a central point, making option A the clear representation of the standard circle equation.

The other options would deviate from this standard definition, either incorrectly altering the signs or modifying the structure of the equation, thus not accurately representing a circle.