# Parametric Equations of an Ellipse

### Parametric Equations of an Ellipse

#### Definition:

Parametric equations are a way to represent the coordinates of points on a curve using a third variable, often denoted as a parameter. For an ellipse, parametric equations express the $x$ and $y$ coordinates of points on the ellipse in terms of a parameter.

#### Parametric Equations for an Ellipse:

The parametric equations for an ellipse centered at $\left(h,k\right)$ with semi-major axis $a$ along the x-axis and semi-minor axis $b$ along the y-axis are:

$x=h+a\mathrm{cos}\left(\theta \right)$  $y=k+b\mathrm{sin}\left(\theta \right)$

Where:

• $\left(h,k\right)$ represents the center of the ellipse.
• $a$ is the length of the semi-major axis.
• $b$ is the length of the semi-minor axis.
• $\theta$ is the parameter that ranges from $0$ to $2\pi$ to trace the entire ellipse.

#### Interpretation and Properties:

1. Parameter $\theta$:

• Varies from $0$ to $2\pi$ to sweep the ellipse's full path.
• Each value of $\theta$ corresponds to a specific point on the ellipse.
2. Ellipse Points:

• As $\theta$ varies, $x$ and $y$ values change, generating a series of points on the ellipse.
3. Direction of Tracing:

• As $\theta$ increases, the parametric equations trace the ellipse counterclockwise from $0$ to $2\pi$.

#### Special Cases:

1. Circle Parametric Equations:

• For a circle ($a=b$), the parametric equations become: $x=h+r\mathrm{cos}\left(\theta \right)$ $y=k+r\mathrm{sin}\left(\theta \right)$ Where $r$ is the radius of the circle.
2. Changing Parameter Limits:

• Changing the limits of $\theta$ affects how much of the ellipse is traced. For instance, $\theta$ ranging from $0$ to $\pi$ traces only half of the ellipse.

### Example:

Given: An ellipse has a center at $\left(3,-2\right)$, a semi-major axis of length $6$, and a semi-minor axis of length $4$. Find the parametric equations of this ellipse.

#### Solution:

Given: Center of the ellipse: $\left(h,k\right)=\left(3,-2\right)$ Length of the semi-major axis: $a=6$ Length of the semi-minor axis: $b=4$

The parametric equations for an ellipse centered at $\left(h,k\right)$ with semi-major axis $a$ and semi-minor axis $b$ are: $x=h+a\mathrm{cos}\left(\theta \right)$ $y=k+b\mathrm{sin}\left(\theta \right)$

Substitute the given values: $x=3+6\mathrm{cos}\left(\theta \right)$ $y=-2+4\mathrm{sin}\left(\theta \right)$

These equations represent the points on the ellipse in terms of a parameter $\theta$ ranging from $0$ to $2\pi$.

### Conclusion:

The parametric equations for the given ellipse centered at $\left(3,-2\right)$ with a semi-major axis of length $6$ and a semi-minor axis of length $4$ are $x=3+6\mathrm{cos}\left(\theta \right)$ and $y=-2+4\mathrm{sin}\left(\theta \right)$. These equations define the coordinates of points on the ellipse as $\theta$ varies from $0$ to $2\pi$.