Circle Equations

Definition

Algebraic representations of a circle, the set of all points in a plane equidistant from a fixed center $(h, k)$ with radius $r$ .

Equations allow geometric concepts like constant distance to be manipulated algebraically for collision detection and coordinate tracking.

Standard Form: $(x - h)^2 + (y - k)^2 = r^2$ $(x - h)^{2} + (y - k)^{2} = r^{2}$ .
- How to read: “The quantity x minus h squared plus the quantity y minus k squared equals r squared.”
- Meaning / when to use: Direct read-off of center $(h,k)$ and radius $r$ . Derived from the distance formula with fixed distance $r$ .
General Form: $x^2 + y^2 + ax + by + c = 0$ $x^{2} + y^{2} + a x + b y + c = 0$ .
- How to read: “The value x squared plus y squared plus a x plus b y plus c equals zero.”
- Meaning: Expanded form; complete the square on $x$ and $y$ to recover standard form and identify center/radius.
Unit Circle: A circle centered at $(0, 0)$ $(0, 0)$ with $r = 1$ $r = 1$ , defined by $x^2 + y^2 = 1$ $x^{2} + y^{2} = 1$ .
- How to read: “The value x squared plus y squared equals one.”
- Meaning: Every point satisfies $x^2 + y^2 = 1$ ; the foundation for trigonometry on the unit circle.