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

• A. (x - h)^2 + (y - k)^2 = r^2
• B. (x - h)^2 + (y - k)^2 = r
• C. x^2 + y^2 = r^2
• D. (x + h)^2 + (y + k)^2 = r^2

Answer: (A)

The key idea is that a circle consists of all points whose distance from its center (h, k) is a fixed radius r. Using the distance formula, that distance is sqrt[(x − h)^2 + (y − k)^2], and setting it equal to r gives sqrt[(x − h)^2 + (y − k)^2] = r. Squaring both sides yields the standard form (x − h)^2 + (y − k)^2 = r^2.

The minus signs with h and k show that the circle is translated so its center is at (h, k): if you used plus signs, the center would be at (−h, −k). The right-hand side must be r^2 because you squared the distance, so the radius appears squared. If the center is at the origin, this reduces to x^2 + y^2 = r^2.