Ellipse
Ellipse: An Overview
Definition:
 Ellipse: An ellipse is a type of conic section formed by the intersection of a cone and a plane. It's defined as the set of all points in a plane, the sum of whose distances from two fixed points (the foci) is constant.
Key Elements:

Foci (Focus Singular): Two fixed points inside the ellipse, denoted as F1 and F2. The sum of distances from any point on the ellipse to these foci is constant.

Major and Minor Axes:
 Major Axis: The longest diameter passing through the foci, endpoints called vertices.
 Minor Axis: The shortest diameter perpendicular to the major axis, also with endpoints on the ellipse.

Center: The midpoint of the major and minor axes, denoted as (h, k).
Standard Equation:
The standard form of the equation for an ellipse centered at $(h,k)$ with major axis of length $2a$ and minor axis of length $2b$ is:
$\frac{(xh{)}^{2}}{{a}^{2}}+\frac{(yk{)}^{2}}{{b}^{2}}=1$
Rotated Ellipse: An ellipse that is tilted or rotated at an angle $\theta $ with respect to the xaxis. Its equation can be derived through rotation formulas.
Eccentricity:

Eccentricity ($e$): A measure of how much the ellipse deviates from a circle. It's calculated as $e=\frac{c}{a}$, where $c$is the distance from the center to a focus and $a$ is half the length of the major axis.

For an ellipse, $0<e<1$:
 $e=0$ for a circle.
 $0<e<1$ for an ellipse.
 $e=1$ for a parabola (an extreme case of an ellipse).
 $e>1$ for a hyperbola.
Properties and Characteristics:

Symmetry: Ellipses are symmetric about their major and minor axes.
 FociDistance Relationship: The distance between the foci is constant and equal to $2c$, where $c$ is the distance between the center and each focus.
 Vertex: The points where the ellipse intersects the major and minor axes are called vertices.
 Area: The area of an ellipse is given by $A=\pi ab$, where $a$ and $b$ are the semimajor and semiminor axes, respectively

Relationship between Axes and Eccentricity:
 The relationship $b=a\sqrt{1{e}^{2}}$

Parametric Equations:
 Parametric equations of an ellipse centered at $(h,k)$ with semimajor axis $a$ and semiminor axis $b$ are: $x=h+a\mathrm{cos}(\theta )$ $y=k+b\mathrm{sin}(\theta )$

Application in Orbits and Optics:
 Orbits of celestial bodies often follow elliptical paths.
 In optics, ellipses are used in designing lenses, telescopes, and satellite dishes.