# Equation of the Circle in Various Forms

### Equation of the Circle in Various Forms:

#### 1. Standard Form:

• Equation: $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$
• Center and Radius: Center at $\left(h,k\right)$ and radius $r$.
• Explanation: Represents a circle with center $\left(h,k\right)$ and radius $r$.
• Expanded Form:
• ${x}^{2}-2hx+{h}^{2}+{y}^{2}-2ky+{k}^{2}={r}^{2}$
• ${x}^{2}+{y}^{2}+2gx+2fy+c=0$, where $g=-h$, $f=-k$, and $c={h}^{2}+{k}^{2}-{r}^{2}$.

#### 2. General Form:

• Equation: ${x}^{2}+{y}^{2}+2gx+2fy+c=0$
• Parameters: $g,f,$ and $c$ are constants.
• Center and Radius: Center at $\left(-g,-f\right)$ and radius $r=\sqrt{{g}^{2}+{f}^{2}-c}$.
• Explanation: Represents a circle centered at $\left(-g,-f\right)$ and its radius can be deduced from the coefficients.
• General Equation: $A{x}^{2}+A{y}^{2}+Dx+Ey+F=0$
• Conditions: The coefficients $A$ and $B$ should be equal for the equation to represent a circle.
• Center and Radius: The center of the circle is given by $\left(-\frac{D}{2A},-\frac{E}{2A}\right)$, and the radius is $\sqrt{\frac{{D}^{2}+{E}^{2}-4AF}{4{A}^{2}}}$.

#### 3. Parametric Form:

• Equation: $x=h+r\cdot \mathrm{cos}\theta$ and $y=k+r\cdot \mathrm{sin}\theta$
• Parameters: $h,k$ are the coordinates of the center, $r$ is the radius, and $\theta$ varies from $0$ to $2\pi$.
• Explanation: Represents the circle's points by varying the parameter $\theta$ through trigonometric functions.

#### 4. Diameter Form:

• Equation: $\left({x}_{1}-{x}_{2}{\right)}^{2}+\left({y}_{1}-{y}_{2}{\right)}^{2}=\left(2r{\right)}^{2}$
• Parameters: $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ are the endpoints of the diameter, and $r$ is the radius.
• Explanation: Represents a circle using the endpoints of its diameter and the distance formula.
• Equation with Diameter Endpoints: If the endpoints of the diameter are $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$, the equation of the circle is $\left(x-{x}_{1}\right)\left(x-{x}_{2}\right)+\left(y-{y}_{1}\right)\left(y-{y}_{2}\right)=0$.

#### 5. Vector Form:

• Equation: $\mathbf{r}=\mathbf{c}+r\mathbf{u}$
• Parameters: $\mathbf{r}$ is the position vector of any point on the circle, $\mathbf{c}$ is the position vector of the center, $r$ is the radius, and $\mathbf{u}$ is the unit vector representing direction.
• Explanation: Represents the circle using vectors, where the position vector $\mathbf{r}$ moves along the circle using the center $\mathbf{c}$ and the unit vector $\mathbf{u}$.
• Vector Equation: For a circle with center $\mathbf{c}$ and radius $r$, the vector equation is $\mathrm{\parallel }\mathbf{r}-\mathbf{c}{\mathrm{\parallel }}^{2}={r}^{2}$ where $\mathbf{r}$ is any point on the circle.

#### 6. Implicit Form:

• Equation: $A{x}^{2}+A{y}^{2}+Dx+Ey+F=0$
• Parameters: $A,D,E,$ and $F$ are coefficients.
• Explanation: Represents a circle in its implicit form, usually when the equation is given without directly identifying the center and radius.

#### 7. Polar Form:

• Equation: $r=2a\mathrm{cos}\theta$
• Parameters: $r$ is the distance from the origin to any point on the circle, $a$ is a constant radius, and $\theta$ varies from $0$ to $2\pi$.
• Explanation: Represents the circle using polar coordinates, relating the radius $r$ and the angle $\theta$ with a constant $a$.
• Polar Equation:
• $r=2a\mathrm{cos}\left(\theta -\alpha \right)$
• $r=2b\mathrm{sin}\left(\theta -\beta \right)$ where $a,b$ are the lengths of semi-major and semi-minor axes, and $\alpha ,\beta$ are angles with respect to the coordinate axes.