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Warm-Up

Write the standard equation of the circle with

the given radius and center.

1) 9 (0,0)

2) 1 (0,5)

3) 4 (-8,-1)

4) 5 (4,2)

10.4 Ellipses

- What are the major parts of an ellipse?
- What are the standard forms of an ellipse

equation? - How do you find the value of the foci c when

given a b? - How do you graph an ellipse?
- How do you find the equation of an ellipse from a

graph?

Definition of Ellipse

An ellipse is the set of all points P in a plane

such that the sum of the distances from P to two

fixed points, F1 and F2, called the foci, is a

constant.

Standard Equation of an Ellipse

Horizontal Major Axis

a2 gt b2 a2 b2 c2

length of major axis 2a

length of minor axis 2b

Standard Equation of an Ellipse

Vertical Major Axis

a2 gt b2 a2 b2 c2

length of major axis 2a

length of minor axis 2b

Example 1

Write the standard equation for an ellipse with

foci at (-8,0) and (8,0) and with a major axis of

20. Sketch the graph.

length of major axis 2a

2a 20,

so a 10

a2 b2 c2

102 b2 82

b2 100 - 64

b2 36,

so b 6

Example 2

Find the vertices and co-vertices of the ellipse.

vertices

(0,7) and (0,-7)

co-vertices

(4,0) and (-4,0)

Example 3

Write the standard equation of the ellipse.

length of major axis 2a

2a 16,

so a 8

length of minor axis 2b

2b 8,

so b 4

Practice

Write the standard equation for an ellipse with

foci at (5,0) and (-5,0) and with vertices at

(9,0) and (-9,0). Sketch the graph.

- What are the major parts of an ellipse?
- Vertices, co-vertices, foci, center, major

minor axis - What are the standard forms of an ellipse

equation? - How do you find the value of the foci c when

given a b? - c2 a2 b2
- How do you graph an ellipse?
- Plot the vertices, the co-vertices and draw the

ellipse. - How do you find the equation of an ellipse from a

graph? - a vertex of major axis goes into the

denominator under major axis letter, and b

co-vertex of minor axis goes into the denominator

under the minor axis letter in the formula.

Assignment 10.4

p.612, 19-67 Skip 41, 47

Warm-Up

Write the standard equation for an ellipse with

foci at (-5,0) and (5,0) and with a major axis of

18. Sketch the graph.

10.4 Day 2 Ellipses

- What is the standard equation for an ellipse if

the vertex has been translated?

Standard Equation of a Translated Ellipse

Horizontal Major Axis

a2 gt b2 a2 b2 c2

length of major axis 2alength of minor axis 2b

Standard Equation of a Translated Ellipse

Vertical Major Axis

a2 gt b2 a2 b2 c2

length of major axis 2alength of minor axis 2b

Example 1

An ellipse is defined by the equation 4x2 9y2

16x 18y 11. Write the standard equation and

identify the coordinates of the center, vertices,

co-vertices, and foci. Sketch the graph of the

ellipse.

4x2 16x 9y2 18y 11

4(x2 4x) 9(y2 2y) 11

4(x2 4x 4) 9(y2 2y 1) 11 4(4) 9(1)

4(x 2)2 9(y 1)2 36

Example 1

Write an equation of the ellipse with foci at

(3,5) and (3,-1) and vertices at (3,6) and (3,-2).

Plot the given points and make a rough sketch.

The ellipse has a vertical axis, so its equation

is in the form of

Find the center which is halfway between the

vertices.

Find a and c a 6-24 c 5-23

Find b using b2 a2-c2

- What is the standard equation for an ellipse if

the vertex has been translated?

Assignment 10.4 day 2

p. 612, 24-60 even p. 628, 17, 18, 24, 25