What is the standard form of the equation of a circle with center (h,k) and radius r?

What is the standard form of the equation of a circle with center (h,k) and radius r?

• A. (x - h)^2 + (y - k)^2 = r^2
• B. x^2 + y^2 = 25
• C. (x + h)^2 + (y + k)^2 = r^2
• D. x^2 - y^2 = r^2

Answer: (A)

A circle centered at (h, k) with radius r is described by the equation (x − h)^2 + (y − k)^2 = r^2. This comes from the distance from a point (x, y) to the center (h, k) being r, so the distance formula gives sqrt((x − h)^2 + (y − k)^2) = r, and squaring both sides yields the standard form. This form makes the center and the radius immediately visible: the center is (h, k) and the radius is r. Expanding confirms there’s the same coefficient for x^2 and y^2 and no xy term, which is a hallmark of a circle. Other forms don’t match the general description. A specific circle with center at the origin and radius 5 would be a valid circle, but it fixes the values of h, k, and r rather than representing the general case. A form with (x + h)^2 + (y + k)^2 = r^2 would place the center at (−h, −k), not (h, k). An equation like x^2 − y^2 = r^2 describes a hyperbola, not a circle.

A circle centered at (h, k) with radius r is described by the equation (x − h)^2 + (y − k)^2 = r^2. This comes from the distance from a point (x, y) to the center (h, k) being r, so the distance formula gives sqrt((x − h)^2 + (y − k)^2) = r, and squaring both sides yields the standard form. This form makes the center and the radius immediately visible: the center is (h, k) and the radius is r. Expanding confirms there’s the same coefficient for x^2 and y^2 and no xy term, which is a hallmark of a circle.

Other forms don’t match the general description. A specific circle with center at the origin and radius 5 would be a valid circle, but it fixes the values of h, k, and r rather than representing the general case. A form with (x + h)^2 + (y + k)^2 = r^2 would place the center at (−h, −k), not (h, k). An equation like x^2 − y^2 = r^2 describes a hyperbola, not a circle.