# Equation of the Circle in Various Forms

### Equation of the Circle in Various Forms:

#### 1. **Standard Form:**

**Equation:**$(x-h{)}^{2}+(y-k{)}^{2}={r}^{2}$**Center and Radius:**Center at $(h,k)$ and radius $r$.**Explanation:**Represents a circle with center $(h,k)$ 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 $(-g,-f)$ and radius $r=\sqrt{{g}^{2}+{f}^{2}-c}$.**Explanation:**Represents a circle centered at $(-g,-f)$ 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 $(-\frac{D}{2A},-\frac{E}{2A})$, 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:**$({x}_{1}-{x}_{2}{)}^{2}+({y}_{1}-{y}_{2}{)}^{2}=(2r{)}^{2}$**Parameters:**$({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ 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 $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$, the equation of the circle is $(x-{x}_{1})(x-{x}_{2})+(y-{y}_{1})(y-{y}_{2})=0$.

#### 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:**- $r=2a\mathrm{cos}(\theta -\alpha )$
- $r=2b\mathrm{sin}(\theta -\beta )$ where $a,b$ are the lengths of semi-major and semi-minor axes, and $\alpha ,\beta $ are angles with respect to the coordinate axes.