Circle Equation

Overview

Engineering reference data for Circle Equation in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Circle Equation Forms

The equation of a circle can be expressed in different forms depending on the given information and application.

Unit Circle

A unit circle has its center at the origin $(0, 0)$ and a radius equal to 1:

$x^2 + y^2 = 1$

Standard Form

The standard form expresses a circle with center $(a, b)$ and radius $r$:

$(x - a)^2 + (y - b)^2 = r^2$

General Form

The general form uses coefficients $A$, $B$, and $C$:

$x^2 + y^2 + Ax + By + C = 0$

Converting Between Forms

The general form coefficients relate to the standard form parameters as follows:

Conversion Example

To convert from general form to standard form, complete the square:

$x^2 + Ax + y^2 + By = -C$

$\left(x + \frac{A}{2}\right)^2 + \left(y + \frac{B}{2}\right)^2 = \frac{A^2}{4} + \frac{B^2}{4} - C$

Properties

• A circle exists only when $a^2 + b^2 - C > 0$
• If $a^2 + b^2 - C = 0$, the equation represents a single point (degenerate circle)
• If $a^2 + b^2 - C < 0$, no real circle exists (empty set)

Unit Circle Property

A unit circle is a special case of the standard form where the center is at the origin $(0, 0)$ and the radius is $1$. This simplifies the equation to:

$x^2 + y^2 = 1$

This fundamental equation is critical in trigonometry and defining the sine and cosine functions.

Deriving Circle Properties from General Form

Given the general form equation $x^2 + y2 + Ax + By + C = 0$, you can extract key properties.

The radius $r$ is calculated as: $r = \sqrt{a^2 + b^2 - C}$

The coordinates $(a, b)$ of the center are: $a = -\frac{A}{2}$ $b = -\frac{B}{2}$

These relationships allow you to convert a general form equation into its geometric meaning.

References

• Original Source