Lesson 2: Circles

Property

A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center. The equation for a circle of radius $r$ centered at the point $(h, k)$ is

Examples

• The equation for a circle with its center at $(3, -1)$ and a radius of $5$ is $(x - 3)^2 + (y - (-1))^2 = 5^2$, which simplifies to $(x - 3)^2 + (y + 1)^2 = 25$.

Property

To convert a circle equation from general form to standard form, we complete the square for both $x$ and $y$ terms. For an expression $x^2 + bx$, we add and subtract the constant $(\frac{b}{2})^2$ to create a perfect square.

1. Group the $x$ terms and $y$ terms separately
2. For $x^2 + bx$, add and subtract $(\frac{b}{2})^2$
3. For $y^2 + dy$, add and subtract $(\frac{d}{2})^2$
4. Factor each group: $x^2 + bx + (\frac{b}{2})^2 = (x + \frac{b}{2})^2$

Property

To find a circle equation through three non-collinear points, substitute each point into the general form $x^2 + y^2 + Dx + Ey + F = 0$ to create a system of three linear equations in variables $D$, $E$, and $F$.

Examples