What is the standard equation of a circle?

The standard equation of a circle is derived from the definition that a circle is the set of all points in a plane that are a fixed distance (the radius) from a central point (the center of the circle). In a two-dimensional Cartesian coordinate system, if a circle is centered at the point (h, k) with a radius of r, the standard form of its equation is written as:

((x - h)^2 + (y - k)^2 = r^2).

This formulation captures the geometric idea that any point (x, y) on the circle will satisfy the distance criterion from the center (h, k). The squared terms represent the deviation of the point (x, y) from the center's coordinates in the horizontal (x) and vertical (y) directions.

The placement of h and k as negative signs in the equation reflects that moving right from the center increases x-values (therefore x is subtracted from h), and moving up increases y-values (where y is subtracted from k). Hence, the equation correctly represents all points whose distance to the center (h, k) equals the radius r.

This relationship is foundational in coordinate geometry and essential for problems involving circles, demonstrating why this is