Standard Equation of an Ellipse
Standard Equation of an Ellipse
Definition:
The standard equation of an ellipse is a mathematical representation of an ellipse on a Cartesian plane in a specific form that provides essential information about its major and minor axes, center, foci, and eccentricity.
Standard Form:
For an ellipse centered at $(h,k)$ with major axis $2a$ along the xaxis and minor axis $2b$ along the yaxis, the standard equation is:
$\frac{(xh{)}^{2}}{{a}^{2}}+\frac{(yk{)}^{2}}{{b}^{2}}=1$
Where:
 $(h,k)$ represents the center of the ellipse.
 $a$ is the length of the semimajor axis (half of the major axis length).
 $b$ is the length of the semiminor axis (half of the minor axis length).
Elements in the Standard Equation:

Center: Denoted as $(h,k)$, representing the coordinates of the center of the ellipse.

Major Axis (along xaxis): The longest diameter of the ellipse, $2a$.

Minor Axis (along yaxis): The shortest diameter perpendicular to the major axis, $2b$.

Foci: The distances from the center to the foci are $c$ units, where ${c}^{2}={a}^{2}{b}^{2}$.

Eccentricity: Denoted as $e=\frac{c}{a}$, measures the elongation of the ellipse. It ranges from 0 (for a circle) to 1.
Types of Ellipses:

Standard Form Ellipse: The equation is in the standard form centered at $(h,k)$. $\frac{(xh{)}^{2}}{{a}^{2}}+\frac{(yk{)}^{2}}{{b}^{2}}=1$

General Equation Ellipse: An ellipse not necessarily centered at the origin. $\frac{(xh{)}^{2}}{{a}^{2}}+\frac{(yk{)}^{2}}{{b}^{2}}=1$

Rotated Ellipse: An ellipse that is tilted at an angle $\theta $ with respect to the xaxis. Its equation can be derived using rotation formulas.
Special Cases:

Circle: When $a=b$, the ellipse becomes a circle, and the equation simplifies to $\frac{(xh{)}^{2}}{{r}^{2}}+\frac{(yk{)}^{2}}{{r}^{2}}=1$ (where $r=a=b$).

Degenerate Case: If $a=0$ or $b=0$, the ellipse degenerates into a line segment or a single point.
Derivation:
The standard equation of an ellipse can be derived from the geometric definition of an ellipse, emphasizing the constant ratio of distances from any point on the ellipse to the foci.
Transformations and Rotation:

Translation: Shifting the center of the ellipse by adding or subtracting constants within the equation.

Stretching/Compression: Altering the values of $a$ and $b$ in the equation to modify the major and minor axes.

Rotation: Rotating the ellipse about its center by introducing trigonometric functions within the equation.
Example:
Given: An ellipse has a center at $(2,3)$, major axis length $10$ along the xaxis, and minor axis length $6$ along the yaxis. Find the standard equation of this ellipse.
Solution:
Given: Center of the ellipse: $(h,k)=(2,3)$ Length of the semimajor axis: $a=\frac{10}{2}=5$ Length of the semiminor axis: $b=\frac{6}{2}=3$
The standard equation of an ellipse with center $(h,k)$, and major and minor axes lengths $2a$ and $2b$ respectively is:
$\frac{(xh{)}^{2}}{{a}^{2}}+\frac{(yk{)}^{2}}{{b}^{2}}=1$
Substituting the given values into the equation:
$\frac{(x2{)}^{2}}{{5}^{2}}+\frac{(y+3{)}^{2}}{{3}^{2}}=1$
This equation represents an ellipse with its center at $(2,3)$, a major axis of length $10$ along the xaxis, and a minor axis of length $6$ along the yaxis.
Conclusion:
The standard equation of the given ellipse is $\frac{(x2{)}^{2}}{25}+\frac{(y+3{)}^{2}}{9}=1$. This equation describes the geometric properties of the ellipse, including its center, major and minor axes lengths, and shape.