homework, page 55

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Homework, Page 55

Find an equation for each circle.

1.Center (–2, 3); radius 3

2 22 3 9x y

2 22 3 9x y

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Homework, Page 55

Find an equation for each circle.

3.Center (0, 3); radius 12

2 2 20 3 12x y

22 3 144x y

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Homework, Page 55

Graph, if possible. Find center and radius.

5. 132 22 yx

: 2,3 ; 1Center r

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Homework, Page 55

Graph, if possible. Find center and radius.

7. 2 24 4x y

: 4,0 ; 2Center r

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Homework, Page 55

Graph, if possible. Find center and radius.

9. 0222 yyx

2 2 2 0x y y

2 2 2 1 1x y y

22 1 1x y : 0,1 ; 1Center r

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Homework, Page 55Graph, if possible. Find center and radius.

11. 2 2 5 7 0x y x y

2 2 5 7 0x y x y 2 25 7 0x x y y

2 25 6.25 7 12.5 6.25 12.25x x y y

2 22.5 3.5 18.5x y

: 2.5, 3.5 ; 4.301Center r

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Homework, Page 55

Graph, if possible. Find center and radius.13.

Empty graph.

72 22 yx

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Homework, Page 55

Graph, if possible. Find center and radius.15.

2 2 8 16x y x

2 2 8 16x y x 2 28 16x x y

2 28 16 16 16x x y

2 24 0x y

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Homework, Page 55

Find an equation of the line tangent to the circle at P.

17. 2 2( 2) 10; (1, 3); (2,0)x y P C

0 33

2 1m

13 1

3y x

3 9 1y x

3 8x y

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Homework, Page 55

Determine if A is inside, on, or outside the circle.

19. C = (2, –1); r = 3; A = (3, 2)

2 22 1 9x y

2 23 2 2 1 9

10 9 Outside the circle

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Homework, Page 55

Determine if A is inside, on, or outside the circle.

21. C = (0, 0); r = 4; A = (2, 2) 2 2 16x y 2 22 2 16

8 16 Inside the circle

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Homework, Page 55

Find an equation of each circle.

23. Center (3, 5); tangent to the x-axis

2 23 5 25x y

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Homework, Page 55

Find an equation of each circle.

25. Tangent to the x-axis, the y-axis, and the line y = 5. (two answers)

2 22.5 2.5 6.25x y

2 22.5 2.5 6.25x y

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Homework, Page 55

Find an equation of each circle.

27. Center on the line y = 1 – 2x, tangent to the y-axis at (0, 3)

1 2y x

x

y

3 1 2x 1x

2 21 3 1x y

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Homework, Page 55

29. Find an equation of the circle containing (–9, 2), (–1, 2), (–1, 6), and (–9, 6)

9 1 6 2, 5,4

2 2M M

x

y

2 29 5 6 4 16 4 20d

2 25 4 20x y 2 2

5 4 20x y

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8.1

Conic Sections and Parabolas

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What you’ll learn about

Conic Sections Geometry of a Parabola Translations of Parabolas Reflective Property of a Parabola

… and whyConic sections are the paths of nature: Any free-moving object in a gravitational field follows the path of a conic section.

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A Right Circular Cone (of two nappes)

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Conic Sections

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Degenerate Conic Sections

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Second-Degree (Quadratic) Equations in Two Variables

2 2 0,

where , , and , are not all zero.

Ax Bxy Cy Dx Ey F

A B C

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Parabola

A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.

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Parabolas with Vertex (0,0)

Stand ard eq uation x2 = 4py y2 = 4pxOp en s Upward or To the right or to the downwar d left

Fo cu s (0 ,p) (p,0 )Directrix y = –p x = –pAx is y-ax isx-ax isFo cal len gth p pFo cal width |4 p | |4 p |

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Graphs of x2=4py

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Graphs of y2 = 4px

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Example Finding an Equation of a Parabola

Find an equation in standard form for the

parabola whose directrix is the line 3

and whose focus is the point ( 3,0).

x

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Parabolas with Vertex (h,k)

Stand ard eq uation (x–h)2 = 4p(y–k) (y–k)2 = 4p(x–h)

Op en s Upward or To the right or to the downwar d left

Fo cu s (h,k+p) (h+p,k)Directrix y = k–p x = h–p Ax is x = h y = kFo cal len gth p pFo cal width |4 p | |4 p |

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Example Finding an Equation of a Parabola

Find the standard form of the equation for

the parabola with vertex at (1,2) and focus

at (1, 2).

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Example Graphing a Parabola with a Calculator

2

Graph the parabola given by: 4 4 2 .y x

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Example Solving a word Problem About Parabolas

62. Stein Glass, Inc. makes parabolic headlights for a variety of automobiles. If one of its headlights has a parabolic surface generated by the parabola x2 = 12y, where should the light bulb be placed?

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Homework

Homework Assignment #21 Review Section: 7.1 Page 641, Exercises: 1 – 69 (EOO)

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8.2

Ellipses

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Quick Review

2 2

2 2

2

1. Find the distance between ( , ) and (1,2).

2. Solve for in terms of . 19 4

Solve for algebraically.

3. 3 8 3 12 10

4. 6 1 6 12 11

5. Find the exact solution by completing the square.

2

a b

y xy x

x

x x

x x

x

8 21 0x

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Quick Review Solutions

2 2

2

2

2

2

2

1 2

3

1. Find the distance between ( , ) and (1,2).

2. Solve for in terms of . 1 9 4

Solve for algebraically.

3. 3 8 3 1

6 9

2

82 10

4. 6 1 6 12 11

5. Find the exact solut

2

a b a b

xy

x

x

y xy x

x

x x

x x

2

ion by completing the square.

2 8 21 0 29

22

xx x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 35

What you’ll learn about

Geometry of an Ellipse Translations of Ellipses Orbits and Eccentricity Reflective Property of an Ellipse

… and whyEllipses are the paths of planets and comets around the Sun, or of moons around planets.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 36

Ellipse

An ellipse is the set of all points in a plane whose distance from two fixed points in the plane have a constant sum. The fixed points are the foci (plural of focus) of the ellipse. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the ellipse intersects its axis are the vertices of the ellipse.

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Key Points on the Focal Axis of an Ellipse

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Ellipse - Additional Terms

The major axis is the chord connecting the vertices of the ellipse. The semimajor axis is the distance from the center of the ellipse and to one of the vertices. The minor axis is the chord perpendicular to the major axis and passing through the center of the ellipse. The semiminor axis is the distance from the center of the ellipse to one end of the minor axis, sometimes called a minor vertex.

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Ellipse with Center (0,0)

2 2 2 2

2 2 2 2 Standard equation 1 1

Focal axis -axis -axis

Foci ( ,0)

x y y x

a b a bx y

c

(0, )

Vertices ( ,0) (0, )

Semimajor axis

Semiminor axis

c

a a

a a

b b

2 2 2 2 2 2 Pythagorean relation a b c a b c

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Pythagorean Relation

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Example Finding the Vertices and Foci of an Ellipse

2 2Find the vertices and the foci of the ellipse 9 4 36.x y

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Example Finding an Equation of an Ellipse

Find an equation of the ellipse with foci ( 2,0) and (2,0) whose minor

axis has length 2.

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Ellipse with Center (h,k)

2 2 2 2

2 2 2 2 Standard equation 1 1

Focal axis

Foci ( , )

x h y k y k x h

a b a by k x h

h c k

( , )

Vertices ( , ) ( , )

Semimajor axis

Semiminor ax

h k c

h a k h k a

a a

2 2 2 2 2 2

is

Pythagorean relation

b b

a b c a b c

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Ellipse with Center (h,k)

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Example Locating Key Points of an Ellipse

2 2

Find the center, vertices, and foci of the ellipse

1 11

4 9

x y

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Example Finding the Equation of an Ellipse

Find the equation of the ellipse with major axis endpoints 3, 7 and 3,3

and minor axis length 6.

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Example Graphing an Ellipse Using Parametric Equations

2 2

1 2Parameterize and graph the ellipse whose equation is: 1

16 9

y x

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Example Proving an Ellipse

2 2

Prove that the graph of the equation is an ellipse, and find its center,

vertices, and foci: 9 16 54 32 47 0x y x y

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Elliptical Orbits Around the Sun

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Eccentricity of an Ellipse

2 2

The of an ellipse is ,

where is the semimajor axis, is the semiminor

axis, and is the distance from the center of the

ellipse to either focus.

c a be

a aa b

c

eccentricity