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 $(0,0)$. However, ellipses might not always be centered at the origin. The general equation of an ellipse with its center at $(h,k)$ can be expressed as:
$\frac{(xh{)}^{2}}{{a}^{2}}+\frac{(yk{)}^{2}}{{b}^{2}}=1$
Where:
 $(h,k)$ denotes the coordinates of the center.
 $a$ is the length of the semimajor axis.
 $b$ is the length of the semiminor 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 $(h,k)$, semimajor axis $a$, and semiminor axis $b$ is given by:
$x=h+a\cdot \mathrm{cos}(t)$$y=k+b\cdot \mathrm{sin}(t)$
Where $t$ varies over the interval $[0,2\pi ]$ for a complete ellipse.
Rotated Ellipse Equation:
An ellipse can be rotated at an angle $\theta $ with respect to the xaxis. The equation of a rotated ellipse, centered at $(h,k)$, can be derived using rotation formulas. The general form of a rotated ellipse is:
$\frac{(xh{)}^{2}}{{a}^{2}\cdot {\mathrm{cos}}^{2}(\theta )}+\frac{(yk{)}^{2}}{{b}^{2}\cdot {\mathrm{sin}}^{2}(\theta )}=1$
Where $a$ and $b$ are the semimajor and semiminor 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 $(0,0)$ with semimajor axis $a$ and semiminor 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 $(0,0)$.
 It's a more generalized form of the ellipse equation.
Polar Equation of Ellipse:
$r(\theta )=\frac{ab}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}(\theta )+{b}^{2}{\mathrm{cos}}^{2}(\theta )}}$
 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 xaxis and minor axis along the yaxis.
 $a$ and $b$ represent the semimajor and semiminor axes, respectively.
Transformations and Rotations:

Horizontal and Vertical Translations:
 Adding or subtracting constants to $x$ and $y$ terms in the standard equation shifts the ellipse horizontally or vertically.

Rotation of Axes:
 An ellipse can be rotated by an angle $\theta $ by applying rotation formulas to $x$ and $y$ in the standard equation.

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 $3{x}^{2}+4{y}^{2}6x+16y11=0$. Find the center, semimajor and semiminor 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 $x$: $3({x}^{2}2x)+4{y}^{2}+16y=11$ $3({x}^{2}2x+1)3+4({y}^{2}+4y+4)16=11$
$3(x1{)}^{2}+4(y+2{)}^{2}=30$
Divide by $30$ to get the standard form: $\frac{(x1{)}^{2}}{10}+\frac{(y+2{)}^{2}}{7.5}=1$
Now, we have the equation in the standard form. The center is $(h,k)=(1,2)$, the semimajor axis is $a=\sqrt{10}$, and the semiminor axis is $b=\sqrt{7.5}$.
Example 2: Parametric Equations of Ellipse
Given: The center of an ellipse is $(3,2)$, with a semimajor axis $a=4$ and a semiminor axis $b=3$. Find the parametric equations of the ellipse.
Solution:
The parametric equations for an ellipse centered at $(h,k)$ with semimajor axis $a$ and semiminor axis $b$ are:
$x=h+a\mathrm{cos}(\theta )$ $y=k+b\mathrm{sin}(\theta )$
Substitute the given values: $x=3+4\mathrm{cos}(\theta )$ $y=2+3\mathrm{sin}(\theta )$
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 semimajor axis $a=5$ and semiminor axis $b=3$. Find the polar equation of the ellipse.
Solution:
The polar equation for an ellipse centered at the origin with semimajor axis $a$ and semiminor axis $b$ is:
$r(\theta )=\frac{ab}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}(\theta )+{b}^{2}{\mathrm{cos}}^{2}(\theta )}}$
Substitute the given values: $r(\theta )=\frac{5\times 3}{\sqrt{25{\mathrm{sin}}^{2}(\theta )+9{\mathrm{cos}}^{2}(\theta )}}$
This equation describes the ellipse in polar coordinates with respect to the angle $\theta $.