# Equation of Ellipse in Other Forms

### Equation of Ellipse in Other Forms

#### General Equation of an Ellipse:

The standard equation of an ellipse represents an ellipse centered at the origin $\left(0,0\right)$. However, ellipses might not always be centered at the origin. The general equation of an ellipse with its center at $\left(h,k\right)$ can be expressed as:

$\frac{\left(x-h{\right)}^{2}}{{a}^{2}}+\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$

Where:

• $\left(h,k\right)$ denotes the coordinates of the center.
• $a$ is the length of the semi-major axis.
• $b$ is the length of the semi-minor axis.

#### Equation of an Ellipse in Parametric Form:

The parametric equations of an ellipse allow the representation of points on the ellipse using parameters $t$ (usually angle or time). The parametric form of an ellipse with center $\left(h,k\right)$, semi-major axis $a$, and semi-minor axis $b$ is given by:

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

Where $t$ varies over the interval $\left[0,2\pi \right]$ for a complete ellipse.

#### Rotated Ellipse Equation:

An ellipse can be rotated at an angle $\theta$ with respect to the x-axis. The equation of a rotated ellipse, centered at $\left(h,k\right)$, can be derived using rotation formulas. The general form of a rotated ellipse is:

$\frac{\left(x-h{\right)}^{2}}{{a}^{2}\cdot {\mathrm{cos}}^{2}\left(\theta \right)}+\frac{\left(y-k{\right)}^{2}}{{b}^{2}\cdot {\mathrm{sin}}^{2}\left(\theta \right)}=1$

Where $a$ and $b$ are the semi-major and semi-minor axes lengths respectively, and $\theta$ is the angle of rotation.

#### Cartesian Form from Parametric Equations:

The Cartesian form of an ellipse can be derived from its parametric equations by eliminating the parameter $t$. For an ellipse centered at the origin $\left(0,0\right)$ with semi-major axis $a$ and semi-minor axis $b$, the Cartesian equation is:

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

General Form of Ellipse:

$A{x}^{2}+B{y}^{2}+Cx+Dy+E=0$

• This equation represents an ellipse that might not necessarily have its center at the origin $\left(0,0\right)$.
• It's a more generalized form of the ellipse equation.

Polar Equation of Ellipse:

$r\left(\theta \right)=\frac{ab}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}\left(\theta \right)+{b}^{2}{\mathrm{cos}}^{2}\left(\theta \right)}}$

• Describes the ellipse in polar coordinates with the center at the origin.
• $r$ represents the distance from the center to a point on the ellipse at an angle $\theta$.

Canonical Form of Ellipse:

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

• Represents an ellipse centered at the origin with the major axis along the x-axis and minor axis along the y-axis.
• $a$ and $b$ represent the semi-major and semi-minor axes, respectively.

#### Transformations and Rotations:

1. Horizontal and Vertical Translations:

• Adding or subtracting constants to $x$ and $y$ terms in the standard equation shifts the ellipse horizontally or vertically.
2. Rotation of Axes:

• An ellipse can be rotated by an angle $\theta$ by applying rotation formulas to $x$ and $y$ in the standard equation.
3. Completing the Square:

• The standard form can be obtained from the general form by completing the square for both $x$ and $y$ terms.

### Example 1: General Form of Ellipse

Given: The equation of an ellipse is $3{x}^{2}+4{y}^{2}-6x+16y-11=0$. Find the center, semi-major and semi-minor axes, and sketch the ellipse.

#### Solution:

To rewrite the given equation in the standard form, complete the square for both $x$ and $y$ terms:

$3{x}^{2}-6x+4{y}^{2}+16y=11$

Completing the square for $x$: $3\left({x}^{2}-2x\right)+4{y}^{2}+16y=11$ $3\left({x}^{2}-2x+1\right)-3+4\left({y}^{2}+4y+4\right)-16=11$

$3\left(x-1{\right)}^{2}+4\left(y+2{\right)}^{2}=30$

Divide by $30$ to get the standard form: $\frac{\left(x-1{\right)}^{2}}{10}+\frac{\left(y+2{\right)}^{2}}{7.5}=1$

Now, we have the equation in the standard form. The center is $\left(h,k\right)=\left(1,-2\right)$, the semi-major axis is $a=\sqrt{10}$, and the semi-minor axis is $b=\sqrt{7.5}$.

### Example 2: Parametric Equations of Ellipse

Given: The center of an ellipse is $\left(-3,2\right)$, with a semi-major axis $a=4$ and a semi-minor axis $b=3$. Find the parametric equations of the ellipse.

#### Solution:

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

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

### Example 3: Polar Equation of Ellipse

Given: An ellipse has a center at the origin, with semi-major axis $a=5$ and semi-minor axis $b=3$. Find the polar equation of the ellipse.

#### Solution:

The polar equation for an ellipse centered at the origin with semi-major axis $a$ and semi-minor axis $b$ is:

$r\left(\theta \right)=\frac{ab}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}\left(\theta \right)+{b}^{2}{\mathrm{cos}}^{2}\left(\theta \right)}}$

Substitute the given values: $r\left(\theta \right)=\frac{5×3}{\sqrt{25{\mathrm{sin}}^{2}\left(\theta \right)+9{\mathrm{cos}}^{2}\left(\theta \right)}}$

This equation describes the ellipse in polar coordinates with respect to the angle $\theta$.