How can you use the definition of a circle and the distance formula to derive the standard of a circle?

Answer 1

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. Let the center be at the coordinates ((h, k)) and the distance from the center to any point on the circle be (r). Using the distance formula, the distance (d) between the center ((h, k)) and any point ((x, y)) on the circle is given by (d = \sqrt{(x - h)^2 + (y - k)^2}). Setting this distance equal to the radius (r) and squaring both sides leads to the standard form of a circle's equation: ((x - h)^2 + (y - k)^2 = r^2).