Definition:

A circle is a perfectly round shape where all points are equidistant from a single point called the center. In other words, the circle is the locus of a point which always remains equidistant from a fixed point called the center of the circle. This distance from the center to any point on the circle is known as the radius.

1. Standard Form of a Circle Equation

   The standard form of the equation of a circle with center ( ℎ , 𝑘 ) and radius 𝑟 is: $$(x - h)^2 + (y - k)^2 = r^2$$

2. General Form of a Circle Equation

   The general form of a circle's equation is derived from the standard form and expanded: $$x^2 + y^2 + ax + by + c = 0$$ where a , b , and c are constants that can be determined by expanding and simplifying the standard form equation.

3. Parametric Form

   A circle can also be described using parametric equations with a parameter 𝜃 : $$x = h + r \cos(\theta)$$ $$y = k + r \sin(\theta)$$ where 𝜃 ranges from 0 to 2 𝜋 .

4. Polar Coordinates

   In polar coordinates, the equation of a circle with radius 𝑟 and center at the origin is simply: $$r = \text{constant}$$

Examples

1. Standard Form Example:

   A circle with center ( 3 , − 2 ) and radius 4 : $$(x - 3)^2 + (y + 2)^2 = 16$$

2. General Form Example:

   The same circle in general form: $$x^2 + y^2 - 6x + 4y - 3 = 0$$

3. Parametric Form Example:

   A circle with center ( 0 , 0 ) and radius 5 : \[ \begin{align} x &= 5 \cos(\theta) \\ y &= 5 \sin(\theta) \end{align} \]

4. Polar Coordinates Example:

   A circle with radius 5 and center at the origin: $$r = 5$$