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Ellipse

Conics are the curves formed by the intersection of a plane and a double napped cone. Ellipse is one of the four basic conics which are formed when the Plane does not pass through the vertex of the cone.

A point is called the degenerating conic corresponding to an Ellipse, which is formed when the intersecting plane passes through the vertex of the cone. We find the orbits of planets are in the form of Ellipses.
 
The equations of ellipses are used as models to solve many real life problems.

 Ellipse               
     Ellipse Diagram
  Basic Conic
   The Ellipse
  Degenerating Conic
     The Point.

A conic section is defined as the locus of a point (x, y) which moves such that the ratio of its distances from a fixed point and a fixed line is a constant. The constant is called the Eccentricity of the Ellipse and is represented by 'e'.The fixed point is called the Focus and the fixed line is called the Directrix of the Conic.
The conic formed is an Ellipse when 0 < e < 1.

An Ellipse has two Foci corresponding to two directirces.
An Ellipse is also generally defined as the locus of a point (x, y) which moves such that the sum of its distances from two fixed points (Foci) is always a constant. 

Related Calculators
Ellipse Calculator Perimeter of an Ellipse
Area of a Ellipse Calculator
 

Ellipse Equations

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The equation of an ellipse with center at the origin can be written in two forms based on its orientation. 

The parent Ellipse is a closed curve symmetric about both x and y axis.

The lengths intercepted on the two axes are called the major and the minor axes of the Ellipse, the longer being the major and the shorter the minor.

Ellipse with Horizontal Major Axis
The equation of the ellipse with center at (0, 0) and the major axis along the x axis is,
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$=1            where a > b

Ellipse Equations

The length of the major axis = AA' = 2a
The length of the minor axis = BB' = 2b
The Vertices are A (a, 0) and A' (-a, 0). The covertices are B (0, b) and B' (0, -b).
The two Foci have the coordinates C (c, 0) and C' (-c, 0)
where c2 = a2 - b2.

For example, if the equation of the Ellipse is
$\frac{x^{2}}{25}+\frac{y^{2}}{16}$=1
a = 5 and b = 4
Hence the length of the major axis = 10 and the length of the minor axis = 8.
The focal length c = $\sqrt{a^{2}-b^{2}}$ = $\sqrt{25-16}$ = $\sqrt{9}$ = 3.
The vertices are (5, 0) and (-5, 0). The covertices are (0, 4) and (0, -4).
The foci are (3, 0) and (-3, 0).

Ellipse with Vertical Major Axis
The equation of the ellipse with center (0, 0) and the major axis along the y axis is,
$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}$=1            where a > b

Equation of Ellipse

The length of the major axis = BB' = 2a
The length of the minor axis AA' = 2b
The vertices are B (0, a) and B' (0, -a). The covertices are A (b, 0) and A' (-b, 0).
The two Foci are C (0, c) and C' (0, -c) and c2 = a2 - b2.

For example, consider the equation $\frac{x^{2}}{36}+\frac{y^{2}}{64}$=1
a = 8 and b = 6
The length of the major axis = 16 and the length of the minor axis = 12
The Focal length  c = $\sqrt{a^{2}-b^{2}}$ = $\sqrt{64-36}$ = $\sqrt{28}$ = $2\sqrt{7}$
The vertices are (0, 8) and (0, -8). The covertices are (6, 0) and (-6, 0).
The foci are ( (0, $2\sqrt{7}$)  and (0, -$2\sqrt{7}$).

The eccentricity  e of an ellipse describes its shape and e = $\frac{c}{a}$

The descriptions of Ellipses with center translated to (h, k) are as follows:
Horizontal Orientation
$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}$=1      where a > b     and c2 = a2 - b2.
Center (h, k).
Foci ( h ± c, k)
Major axis: y = k
Vertices (h ± a, k)
Minor axis: x = h
Covertices (h, k ± b).

Vertical Orientation
$\frac{(x-h)^{2}}{b^{2}}+\frac{(y-k)^{2}}{a^{2}}$=1       where a > b     and c2 = a2 - b2.
Center (h, k)
Foci (h, k ± c)
Major axis;  x = h
Verrices ( h, k ± a)
Foci (h, k ± c)
Minor axis: y = k
Covertices ( h ± b, k).

More topics in  Ellipse
Area of an Ellipse
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