Derive the standard and general equations of a circle from its center and radius.
The standard form of a circle with center (h,k) and radius r is (x−h)²+(y−k)²=r². Expanding gives the general form x²+y²+Dx+Ey+F=0.
Standard Form
(x − h)² + (y − k)² = r²
Complete the square: h = −D/2, k = −E/2, r = √(h²+k²−F).
A circle with radius zero is a degenerate case — just a single point at the center.