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Five-Minute Check (over Lesson 7-1) Then/Now New

Vocabulary Key Concept Standard Forms of

Equations for Ellipses Example 1 Graph

Ellipses Example 2 Write Equations Given

Characteristics Key Concept Eccentricity Example

3 Determine the Eccentricity of an

Ellipse Example 4 Real World Example Use

Eccentricity Key Concept Standard Form of

Equations for Circles Example 5 Determine Types

of Conics

5-Minute Check 1

Write y 2 6y 4x 17 0 in standard form.

Identify the vertex, focus, axis of symmetry, and

directrix.

A. (y 3)2 4(x 2) vertex (2, 3) focus

(3, 3) axis of symmetry y 3 directrix x 1

B. (y 3)2 4(x 2) vertex (2, 3) focus

(1, 3) axis of symmetry y 3 directrix x

3 C. (y 3)2 4(x 2) vertex (3, 2) focus

(3, 3) axis of symmetry y 3 directrix x 1

D. (y 3)2 4(x 2) vertex (2, 3) focus

(6, 3) axis of symmetry y 3 directrix x

2

5-Minute Check 2

Write x 2 8x 4y 8 0 in standard form.

Identify the vertex, focus, axis of symmetry, and

directrix.

A. (x 4)2 4(y 2) vertex (4, 2) focus

(3, 2) axis of symmetry x 2 directrix y

5 B. (x 4)2 4(y 2) vertex (4, 2)

focus (4, 1) axis of symmetry x 4

directrix y 3 C. (x 4)2 4(y 2)

vertex (4, 2) focus (4, 3) axis of

symmetry x 4 directrix y 1 D. (x 4)2

4(y 2) vertex (4, 2) focus (4, 2)

axis of symmetry x 4 directrix y 6

5-Minute Check 3

Write an equation for a parabola with focus F (2,

5) and vertex V (2, 3).

A. (x 2)2 8(y 5) B. (x 2)2 8(y 3)

C. (x 2)2 2(y 3) D. (x 2)2 8(y 3)

5-Minute Check 4

Write an equation for a parabola with focus F (2,

2) and vertex V (1, 2).

A. (x 2)2 12(y 2) B. (y 2)2 12(x 2)

C. (y 2)2 12(x 1) D. (x 1)2 12(y 2)

5-Minute Check 5

Which of the following equations represents a

parabola with focus (3, 7) and vertex (-3, 2)?

A. (x 3)2 5(y 2) B. (y 3)2 5(x 2)

C. (x 3)2 20(y 2) D. (y 2)2 20(x 3)

Then/Now

You analyzed and graphed parabolas. (Lesson 71)

- Analyze and graph equations of ellipses and

circles.

- Use equations to identify ellipses and circles.

Vocabulary

- ellipse
- foci
- major axis
- center
- minor axis
- vertices
- co-vertices
- eccentricity

Key Concept 1

Example 1

Graph Ellipses

Example 1

Graph Ellipses

Example 1

Graph Ellipses

Graph the center, vertices, and axes. Then make a

table of values to sketch the ellipse.

Answer

xxx-new art (graph)

Example 1

Graph Ellipses

B. Graph the ellipse 4x 2 24x y 2 10y 3

0.

First, write the equation in standard form.

4x 2 24x y 2 10y 3 0 Original

equation (4x2 24x) (y 2 10y) 3 Isolate

and group like terms. 4(x 2 6x) (y 2

10y) 3 Factor. 4(x 2 6x 9) (y 2 10y

25) 3 4(9) 25 Complete

the squares. 4(x 3)2 (y 5)2 64 Factor

and simplify.

Example 1

Graph Ellipses

Divide each side by 64.

Example 1

Graph Ellipses

Example 1

Graph Ellipses

Graph the center, vertices, foci, and axes. Then

make a table of values to sketch the ellipse.

Answer

Example 1

Graph the ellipse 144x 2 1152x 25y 2 300y

396 0.

Example 2

Write Equations Given Characteristics

A. Write an equation for an ellipse with a major

axis from (5, 2) to (1, 2) and a minor axis

from (2, 0) to (2, 4).

Use the major and minor axes to determine a and b.

The center of the ellipse is at the midpoint of

the major axis.

Example 2

Write Equations Given Characteristics

Midpoint formula

(2, 2) Simplify.

Example 2

Write Equations Given Characteristics

B. Write an equation for an ellipse with vertices

at (3, 4) and (3, 6) and foci at (3, 4) and (3,

2)

The length of the major axis, 2a, is the distance

between the vertices.

Distance formula

a 5 Simplify.

Example 2

Write Equations Given Characteristics

2c represents the distance between the foci.

Distance formula

c 3 Simplify. Find the value of b. c2 a2

b2 Equation relating a, b, and c 32 52

b2 a 5 and c 3 b 4 Simplify.

Example 2

Write Equations Given Characteristics

The vertices are equidistant from the center.

Midpoint formula

(3, 1) Simplify.

Example 2

Write Equations Given Characteristics

Example 2

Write an equation for an ellipse with co-vertices

(8, 6) and (4, 6) and major axis of length 18.

Key Concept 2

Example 3

Determine the Eccentricity of an Ellipse

First, determine the value of c. c2 a2

b2 Equation relating a, b, and c c2 64

36 a2 64 and b2 36

Example 3

Determine the Eccentricity of an Ellipse

Use the values of c and a to find the

eccentricity.

Eccentricity equation

The eccentricity of the ellipse is about 0.66.

Answer about 0.66

Example 3

Determine the eccentricity of the ellipse given

by 36x 2 144x 49y 2 98y 1571.

A. 0.27 B. 0.36 C. 0.52 D. 0.60

Example 4

Use Eccentricity

ASTRONOMY The eccentricity of the orbit of

Uranus is 0.47. Its orbit around the Sun has a

major axis length of 38.36 AU (astronomical

units). What is the length of the minor axis of

the orbit?

The major axis is 38.36, so a 19.18. Use the

eccentricity to find the value of c.

Definition of eccentricity

e 0.47, a 19.18

9.0146 c Multiply.

Example 4

Use Eccentricity

Use the values of c and a to determine b.

c2 a2 b2 Equation relating a, b, and

c. 9.01462 19.182 b2 c 9.0146, a

19.18 33.86 b Multiply.

Answer 33.86

Example 4

PARKS A lake in a park is elliptically-shaped.

If the length of the lake is 2500 meters and the

width is 1500 meters, find the eccentricity of

the lake.

A. 0.2 B. 0.4 C. 0.6 D. 0.8

Key Concept 3

Example 5

Determine Types of Conics

A. Write 9x 2 4y 2 8y 32 0 in standard

form. Identify the related conic.

9x 2 4y 2 8y 32 0 Original equation 9x2

4(y 2 2y) 32 Isolate like terms. 9x2 4(y

2 2y 1) 32 4(1) Complete the square. 9x

2 4(y 1)2 36 Factor and simplify.

Divide each side by 36.

Example 5

Determine Types of Conics

Example 5

Determine Types of Conics

B. Write x2 4x 4y 16 0 in standard form.

Identify the related conic selection.

x 2 4x 4y 16 0 Original equation x 2

4x 4 4y 16 0 4 Complete the square. (x

2)2 4y 16 4 Factor and simplify. (x

2)2 4y 12 Add 4y 16 to each side. (x

2)2 4(y 3) Factor. Because only one term is

squared, the conic selection is a parabola.

Answer (x 2)2 4(y 3) parabola

Example 5

Determine Types of Conics

C. Write x 2 y 2 2x 6y 6 0 in standard

form. Identify the related conic.

x 2 y 2 2x 6y 6 0 Original equation x

2 2x y 2 6y 6 Isolate like terms. x 2

2x 1 y 2 6y 9 6 1 9 Complete

the square. (x 1)2 (y 3)2 16 Factor

and simplify. Because the equation is of the

form (x h)2 (y k)2 r 2, the conic

selection is a circle.

Answer (x 1)2 (y 3)2 16 circle

Example 5

Write 16x 2 y 2 4y 60 0 in standard form.

Identify the related conic.

End of the Lesson