Which equation represents a circle with center (h, k) and radius r in standard form?

Which equation represents a circle with center (h, k) and radius r in standard form?

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

Answer: (C)

Distance from a point (x, y) to the center (h, k) must equal the radius r. Using the distance formula, that distance is sqrt[(x − h)^2 + (y − k)^2], so setting it equal to r and squaring gives (x − h)^2 + (y − k)^2 = r^2. This form places the center exactly at (h, k) because the deviations from h and k are inside the squared terms with minus signs. If you wrote (x + h) or (y + k), the center would be at (−h, −k). Using r on the right instead of r^2 would still describe the same circle only if you interpret the right side as a squared distance, but standard form always uses r^2 to reflect that the left side is a squared distance. The correct standard form for a circle with center (h, k) and radius r is (x − h)^2 + (y − k)^2 = r^2.

Distance from a point (x, y) to the center (h, k) must equal the radius r. Using the distance formula, that distance is sqrt[(x − h)^2 + (y − k)^2], so setting it equal to r and squaring gives (x − h)^2 + (y − k)^2 = r^2. This form places the center exactly at (h, k) because the deviations from h and k are inside the squared terms with minus signs. If you wrote (x + h) or (y + k), the center would be at (−h, −k). Using r on the right instead of r^2 would still describe the same circle only if you interpret the right side as a squared distance, but standard form always uses r^2 to reflect that the left side is a squared distance. The correct standard form for a circle with center (h, k) and radius r is (x − h)^2 + (y − k)^2 = r^2.