11.1 EllipsesObjectives:1. Define an ellipse.2. Write the equation of an ellipse.3. Identify important characteristics of an ellipse.4. Graph ellipses.

EllipseThe set of all points whose distances from

two fixed points add to the same constant. The two fixed points are called the foci.

Standard Equation of an EllipseCentered at the Origin

12

2

2

2

b

y

a

x

Depending on which denominator is larger the ellipse could be elongated along the x-axis or the y-axis. If a and b are equal, the ellipse becomes a circle. For instance, given a circle of radius 2 centered at the origin, the equation is:

When everything is divided by 4, the equation becomes:

422 yx

144

22

yx

Characteristics of an EllipseImportant

Facts: The foci are within

the ellipse The major axis

always contains the foci and is determined by which denominator is larger

The distance between the foci is 2c

The center is the midpoint of the foci and the midpoint of the vertices.

The distance between the vertices are 2a and 2b.

c2 = a2 – b2

Example #1Write the following ellipse in standard form. Then

graph and label the foci, the vertices, & the major and minor axes.

225259 22 yx

135

1925

225225

22525

2259

2

2

2

2

22

22

yx

yx

yx

416

925

22

bac

Example #1Write the following ellipse in standard form. Then

graph and label the foci, the vertices, & the major and minor axes.

225259 22 yx

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

Vertices: (±5,0)Covertices: (0,±3)Foci: (±4,0)Major Axis: x-axisMinor Axis: y-axis

Example #2Solve the following equation for y and graph it using a

graphing calculator.

100425 22 yx

425100

425100

251004

2

22

22

xy

xy

xy

Example #3AA. Find the equation of an ellipse with vertices

at (±8, 0) and foci at Then sketch its graph using the intercepts.

0,53

4.419

19

6445

853

853

53 ,8

2

2

2 222

22

b

b

b

b

b

ca

1

1964

1198

22

2

2

2

2

yx

yx

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

123456789

–1–2–3–4–5–6–7–8–9

y

Example #3BB. Find the equation of an ellipse with foci on the y-axis.

The major axis has a length of 10 and the minor axis has a length of 9. Then sketch the graph.

The length of the axes tells us the distance between the vertices.

Since the major axis is 10 units long and on the y-axis, the vertices are at (0, ±5).

For the minor axis, the covertices are on the x-axis at (±4.5, 0).

This implies a = 5 and b = 4.5

12581

4

125

481

15

29

155.4

22

22

2

2

2

2

2

2

2

2

yx

yx

yx

yx

Example #3BB. Find the equation of an ellipse with foci on the y-axis.

The major axis has a length of 10 and the minor axis has a length of 9. Then sketch the graph.

1

55.4 2

2

2

2

yx

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

123456789

–1–2–3–4–5–6–7–8–9

y

Example #4A furniture maker has a rectangular block of wood that

measures 37 ½ inches by 25 inches. She wants to cut it to make the largest elliptical table top possible. Find an equation of an ellipse she can use, placing the center at the origin and the major axis on the x-axis. Locate the foci and sketch the graph.

37 ½

25

The length of the major and minor axes are basically given from the dimensions of the rectangular piece of wood. This problems works very similarly to the last example.

Example #4Find an equation of an ellipse she can use, placing the

center at the origin and the major axis on the x-axis. Locate the foci and sketch the graph.

37 ½

25

a = 37.5 ÷ 2 = 18.75

b = 25 ÷ 2 = 12.5

0.143125.195

5.1275.18 22

cc

c

4 8 12 16 20–4–8–12–16–20 x

4

8

12

16

20

–4

–8

–12

–16

–20

y

15.1275.18 2

2

2

2

yx

Foci: (±14,0)

Example #5Sirius Radio has 3 satellites that travel in the

same elliptical orbit around the earth. The length of the major axis of the orbit is 52,000 miles, the length of the minor axis of the orbit is 50,000 miles, and Earth is at one focus. Find the minimum and maximum distances from any one of the three satellites to Earth.

Note:

The maximum distance is at a + c and the minimum distance is at a – c.

Example #5Major axis 52,000 miles; minor axis is 50,000

miles. Find the minimum and maximum distances from any one of the three satellites to Earth.

a = 52,000 ÷ 2 = 26,000

b = 50,000 ÷ 2 = 25,000

7141000,000,51

000,000,625000,000,676

000,25000,26 22

c

milesMinmilesMax

859,187141000,26:141,337141000,26: