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 point. 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.

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.

The length of the major axis is , and the length of the minor axis is . 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

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

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

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.

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

Since

Since

Major intercepts: (0, 3), (0, –3)

Length of major axis:

Minor intercepts: (2, 0), (–2, 0)

Length of minor axis:

Foci:

The graph of this ellipse is shown in Figure 2 .

Figure 2 The graph of Example 9.

Example 2: Graph the following ellipse. Find its major and minor intercepts and its foci.

Write in standard form by dividing each side by 100.

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

Major intercepts: (5, 0), (–5, 0)

Minor intercepts: (0, 2), (0, –2)

Foci:

The graph of this ellipse is shown in Figure 3 .

Figure 3 The graph of Example 10.

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

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

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

Foci: ( h+ c,k), ( h – c,k) with

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

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

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

Foci: ( h + c, k) ( h – c, k) with

Example 3: Graph the following ellipse. Find its center, major and minor intercepts, and foci.

Center: (2, –1)

Major intercepts: (2 + 6, –1) = (8, –1)

(2 – 6, –1) = (–4, –1)

Minor intercepts: (2, –1 + 5) = (2, 4)

(2, –1 – 5) = (2, –6)

Foci:

The graph of this ellipse is shown in Figure 4 .

Figure 4 The graph of Example 11.

Example 4: An ellipse has the following equation.

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

Rearrange terms.

Factor out the coefficient of each of the squared terms.

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

Divide each side by 400.

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

Major intercepts: (–1 + 5, 3) = (4, 3)

(–1 – 5, 3) = (–6, 3)

Minor intercepts: (–1, 3 + 4) = (–1, 7)

(–1, 3 – 4) = (–1, –1)

Foci: (–1 + 3, 3) = (2, 3)

(–1 – 3, 3) = (–4, 3)

The graph of this ellipse is shown in Figure 5 .

Figure 5 The graph of Example 12.

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