## Ellipse – Analytic geometry Formulas

### Ellipse

In analytic geometry, an ellipse is a mathematical equation that, when graphed, resembles an egg. An ellipse has two focal points. The distance apart between the two points is one way of describing a particular ellipse. If the two points come together the ellipses become a circle with the point at its center.

### Ellipse with center at the origin

Ellipse with center at the origin and major axis on the x-axis.

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

Ellipse with center at the origin and major axis on the y-axis.

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

### Ellipse with center at (h, k):

Ellipse with center at (h, k) and major axis parallel to the x-axis.

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

Ellipse with center at (h, k) and major axis parallel to the y-axis.

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

### Example:

What is the standard form equation of the ellipse that has vertices (±8,0) and foci (±5,0)?

The foci are on the x-axis, so the major axis is the x-axis. Thus, the equation will have the form

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

The vertices are (±8,0), so a=8 and a2=64

The foci are (±5,0), so c=5 and c2=25

We know that the vertices and foci are related by the equation c2=a2-b2. solving for b2, we have:

$25=64–{b}^{2}$
${b}^{2}=39$

The equation of the ellipse is

$\frac{{x}^{2}}{64}+\frac{{y}^{2}}{39}=1$

### Center:

The center of the ellipse has coordinates (h,k).

### Major Axis:

The major axis of the ellipse is the longest width across it. For a horizontal ellipse, that axis is parallel to the x-axis. The major axis has length 2a. Its endpoints are the major axis vertices, with coordinates (h±a,k).

### Minor Axis:

The minor axis of the ellipse is the shortest width across it. For a horizontal ellipse, it is parallel to the y-axis. The minor axis has length 2b. Its endpoints are the minor axis vertices, with coordinates (h,k±b).

### Foci:

The foci are two points inside the ellipse that characterize its shape and curvature. For a horizontal ellipse, the foci have coordinates (h±c,k), where the focal length c is given by

${c}^{2}={a}^{2}–{b}^{2}$

### Eccentricity:

All conic sections have an eccentricity value, denoted e. All ellipses have eccentricities in the range 0≤e<1. An eccentricity of zero is the special case where the ellipse becomes a circle. An eccentricity of 1 is a parabola, not an ellipse.

The eccentricity is defined as:

$e=\frac{c}{a}$

### Example:

Find the coordinates of the vertices and foci of 25x2+y2=25

${x}^{2}+\frac{{y}^{2}}{25}=1$

So b=1 and a=5. In this example, the major axis is vertical.

So the vertices are at (0,−5) and (0,5).

To find c, we proceed as before:

$c=\sqrt{{a}^{2}–{b}^{2}}$
$=\sqrt{25–1}$
$=\sqrt{24}$
$c=4.899$

So the foci are at (0,−4.9) and (0,4.9).