Ellipse

An ellipse is the set of points in a plane such that the sum of the distances from two fixed points in that plane stays constant. The two points are each called a focus. The plural of focus is foci. The midpoint of the segment joining the foci is called the center of the ellipse. An ellipse has two axes of symmetry. The longer one is called the major axis, and the shorter one is called the minor axis. The two axes intersect at the center of the ellipse (see Figure 1).

Figure 1. Axes and foci of ellipses.

figure

The equation of an ellipse that is centered at (0, 0) and has its major axis along the x‐axis has the following standard form.

equation

The length of the major axis is 2| a|, and the length of the minor axis is 2| b|. The endpoints of the major axis are ( a, 0) and (– a, 0) and are referred to as the major intercepts. The endpoints of the minor axis are (0, b) and (0, – b) and are referred to as the minor intercepts. If ( c, 0) and (– c, 0) are the locations of the foci, then c can be found using the equation

c 2 = a 2b 2

If an ellipse has its major axis along the y‐axis and is centered at (0, 0), the standard form becomes equation

The endpoints of the major axis become (0, a) and (0, – a). The endpoints of the minor axis become ( b, 0) and (– b, 0). The foci are at (0, c) and (0, – c), with

c 2 = a 2b 2

When an ellipse is written in standard form, the major axis direction is determined by noting which variable has the larger denominator. The major axis either lies along that variable's axis or is parallel to that variable's axis.

Example 1

Graph the following ellipse. Find its major intercepts, length of the major axis, minor intercepts, length of the minor axis, and foci.

equation

This ellipse is centered at (0, 0). Since the larger denominator is with the y variable, the major axis lies along the y‐axis.

equation

Major intercepts: (0, 3), (0, –3)

Length of major axis: 2 | a| = 6

Minor intercepts: (2, 0), (–2, 0)

Length of minor axis: equation

Foci: equation

The graph of this ellipse is shown in Figure 2.

Figure 2. The graph of Example.

figure

Example 2

Graph the following ellipse. Find its major and minor intercepts and its foci.

4 x 2 + 25 y 2 = 100

Write 4 x 2 + 25 y 2 = 100 in standard form by dividing each side by 100.

equation

This ellipse is centered at (0, 0). Since the larger denominator is with the x variable, the major axis lies along the x‐axis.

equation

Major intercepts: (5, 0), (–5, 0)

Minor intercepts: (0, 2), (0, –2)

Foci: equation,  equation

The graph of this ellipse is shown in Figure 3.

The standard form for an ellipse centered at ( h, k) with its major axis parallel to the x‐axis is equation

Major intercepts: ( h + a, k), ( ha, k)

Minor intercepts:  ( h, k + b), ( h, kb)

Foci: ( h + c, k), ( h – c, k) with equation

The standard form for an ellipse centered at ( h, k) with a major axis parallel to the y‐axis is equation

Major intercepts: ( h, k + a) ( h, ka)

Minor intercepts: ( h + b, k) ( hb, k)

Foci: ( h, k + c) ( h, kc) with equation

The points where the major axis intersects the ellipse are also known as the ellipse's vertices. That means that each major intercept is also known as a vertex of the ellipse. Notice that the vertices, foci, and center of an ellipse all have the same horizontal coordinate when the eclipse's major axis is parallel to the y‐axis, and all have the same vertical coordinate when the major axis parallels the x‐axis. Ellipses may have axes at oblique angles as well as horizontal and vertical, but their study is beyond the scope of this course.

Figure 3. The graph of Example.

figure

Example 3

Graph the following ellipse. Find its center, vertices, minor intercepts, and foci.

equation

Center: (2, –1)

equation

Vertices: equation

Minor intercepts: equation

Foci: equation

The graph of this ellipse is shown in Figure 4.

Figure 4. The graph of Example.

figure

Example 4

An ellipse has the following equation.

16 x 2 + 25 y 2 + 32 x – 150 y = 159

Find the coordinates of its center, major and minor intercepts, and foci. Then graph the ellipse.

16 x 2 + 25 y 2 + 32 x – 150 y = 159

Rearrange terms to get the x‐terms together and the y‐terms together for potential factoring.

16 x 2 + 32 x + 25 y 2 – 150 y = 159

Factor out the coefficient of each of the squared terms.

16( x 2 + 2 x) + 25( y 2 – 6 y) = 159

Complete the square within each set of parentheses and add the same amount to both sides of the equation.

equation

Divide each side by 400.

equation

Center (–1, 3): Since the x variable has the larger denominator, the major axis is parallel to the x‐axis.

equation

Major intercepts: equation

Minor intercepts: equation

Foci: equation

The graph of this ellipse is shown in Figure 5.

Ellipses are eccentric, a property that is expressed as a number between 0 and 1. If the eccentricity of an ellipse were 0 (which it cannot be), that ellipse would be a circle. The greater the eccentricity of an ellipse (the closer it is to 1), the more oval in shape the ellipse is. The eccentricity of an ellipse, e, may be calculated as the ratio of c, equation to a, or equation. Consider the last ellipse you graphed,

equation

Figure its eccentricity by the formula, using a = 5 and equation

Then equation

Since e = 0.6, and 0.6 is closer to 1 than it is to 0, the ellipse in question is much more oval than it is round.

Figure 5. The graph of Example.

figure