Chapter 10wellsmat.startlogic.com/sitebuildercontent/site... · Lesson 10-5 Hyperbolas. Uses of Hyperbolas. Definition A hyperbola is the set of points in a plane the difference of

Chapter 10

Exploring Conic Sections

Conics

A conic section is a curve formed by the intersection

of a plane and a hollow cone.

Each of these shapes are made by slicing the cone and

observing the shape of the slice.

Lesson 10-2

Parabolas

Uses of Parabolas

Uses of Parabolas

http://ccins.camosun.bc.ca/~jbritton/paragolf_lg.GIF

Uses of Parabolas

http://ccins.camosun.bc.ca/~jbritton/parasatellite_lg.GIF

Parabola

A parabola is the set

of all points in a plane

that are the same

distance from a fixed

line (directrix) and a

fixed point (focus) not

on the line

Standard Form of a Parabola

Focus:

Directrix:

( , )h k p

y k p

2( ) 4 ( )x h p y k

Vertex: ( , )h k

Standard form

Axis of symmetry is vertical (y-axis)

opens upward0p

0p opens downward

F

V

D

p

p

2 4x py

Standard Form of a Parabola

Focus:

Directrix:

( , )h p k

x h p

2( ) 4 ( )y k p x h

Vertex: ( , )h k

Standard form

Axis of symmetry is horizontal (x-axis)

opens to the right0p

0p opens to the left

2 4y px

Fp p

D

V

Example 5 – Page 547, #24

Identify the vertex, the focus, and the directrix and sketch

the graph.

21

24x y

Step 1 – Locate which variable has the square.

-term; the parabola has horizontal symmetry (x-axis)2y


21

24x y

Step 2 – Rewrite the equation in standard form.

2 4y px

2

1 24

x y

2 24y x


Step 3 – Identify the vertex, focus and directrix.

2 24y x

Vertex

(0,0)

Focus

(0 6,0) (6,0)

Directrix:

0 6

6

x

x

(6,0)

(0,0)

2 24y x

Solve for p

4 24

6

p

p

2 4y xp

6x



the graph.

2( 2) 4x y


-term the parabola has vertical symmetry (y-axis)2x


2( 2) 4x y


2( ) 4 ( )x h p y k

Already in standard form.


2( 2) 4x y

Step 3 – Identify the vertex, focus and directrix.2( 2) 4x y

Vertex

(2,0)

Focus

Solve for p

4 4

1

p

p

(2,0 1) (2,1)

Directrix

0 1

1

y

y

2 4( ) ( )px h y k

1y



the graph.2 24 8 16x y x


-term the parabola has vertical symmetry (y-axis)2x


2 24 8 16x y x

Step 2 – Rewrite the equation in standard form.2( ) 4 ( )x h p y k

Complete the square on the x-terms2 8 24 16x x y

2 24 168x x y

284 16

2

2 16 2 1 64 6 18x x y

2( 4) 24x y


2 4( ) ( )px h y k 2( 4) 24x y


Vertex

(4,0)

Focus

Solve for p

4 24

6

p

p

(4,0 ( 6)) (4, 6)

Directrix

0 ( 6)

6

y

y

2 24( 4)x y 6y

Lesson 10-3

Circles

Circle

A circle is the set of all

points in a plane that

are a distance

r (radius) from a

given point (center)

r

Center

Standard Form of a Circle

22 2( )x h y k r

Vertex: ( , )h k

Standard form 2 2 2x y r

( , )h k

r

Example 5, Page 553, #34

Use the center and the radius to graph the circle

2 2( 4) 144x y

Center:

Radius:

( , ) (0, 4)h k

2 144 12r

22 2( )x h y k r


Use the center and the radius to graph the circle

2 21 ( 3) 16x y

Center:

Radius:

( , ) (1, 3)h k

2 16 4r

22 2( )x h y k r


2 22 1 3 1x x y

Find the center and the radius of the circle.

2 22 1 4x x y


2 22 3x x y

2 2 32x x y

221 1

2

22 2( )x h y k r


2 22 1 3 1x x y

Center:

Radius:

( 1,0)

4 2

2 21 4x y

22 2( )x h y k r

Lesson 10-4

Ellipse

Uses of Ellipses

http://community.webshots.com/photo/5812149/5812460LYfSeOalIu

http://www.scottso.net/examples_encsol.htm

http://www.sciencemuseum.org.uk/on-line/electron/index.asp

Uses of Ellipses

Uses of Ellipses

http://ccins.camosun.bc.ca/~jbritton/elliwhisp_lg.GIF

http://ccins.camosun.bc.ca/~jbritton/ellireflect_lg.GIF

Ellipses

An ellipse can be

defined as the set of all

points (P) in plane the

sum of whose distances

from two fixed points

(F1 and F2) is constant.

These two fixed points

are called the foci.

1F 2F

P

Standard Form of a Ellipse

Endpoints:

Minor Axis

Focus:

( , )

( , )

h k b

h k b

( , )

( , )

h k

h

c

c k

2 2

2 21

x h y k

a b

Vertices:

Major Axis

( , )

( , )

h a k

h a k

Standard form 2 2

2 21

x y

a b

Center: ( , )h k

2 2 2c a b

FV

E

c

V

E

F

c

Standard Form of a Ellipse

Endpoints:

Minor Axis

Focus:

( , )

( , )

h b k

h b k

( , )

( , )c

h k

h

c

k

2 2

2 21

x h y k

b a

Vertices:

Major Axis

( , )

( , )

h k a

h k a

Standard form

2 2

2 21

x y

b a

Center: ( , )h k

2 2 2c a b

F

V

Ec

V

E

F

c


Find the foci for each equation of an ellipse. Then graph

the ellipse.

2 2

14 9

x y

Step 1 – Rewrite the equation in standard form

2 2

2 21

x y

b a

in standard form


2 2

14 9

x y

Step 2 – Identify a² and b² to determine which axis is the

major

2

2

9

4

a

b

major axis is the y

2 2

2 21

x y

b a

Example 3, Page 559, #182 2

14 9

x y

Step 3 – Find the center, vertices, endpoints and focus.

2

2

9

4

a

b

major axis is the y

Center: ( , ) (0,0)h k

Vertices: ( , ) (0,0 3)h k a

Endpoints: ( , ) (0 2,0)h b k

Focus:

2 2 2c a b

2 9 4

5 2.23

c

c

( , ) (0,0 5)h k c


Find the foci and graph the ellipse

2 24 16x y


2 24 16

16 16 16

x y

2 2

116 4

x y

2 2

2 21

x y

a b



major

2

2

16

4

a

b

major axis is the x

2 2

116 4

x y

2 2

2 21

x y

a b


116 4

x y

2

2

16

4

a

b


major axis is the x

Center: ( , ) (0,0)h k

Vertices: ( , ) (0 4,0)h a k

Endpoints: ( , ) (0,0 2)h k b

Focus:

2 2 2c a b 2 16 4

12 2 3

c

c

( , ) (0 2 3,0)h c k


Find the foci of the ellipse

2 22 8 4 0x x y


2 2(2 8 ) 4x x y

2 22( 4 ) 4x x y

2 22 44x x y

24

2 42

2 242 4 4 2 4x x y


2 242 4 4 2 4x x y

2 2

2 2

2 2 4 8

2 2 4

x y

x y

2 22 2 4

4 4 4

x y

2 22

12 4

x y


2 22

12 4

x y


major

2

2

4

2

a

b

2 2 2c a b

Focus: ( , )h k c

2 4 2

2

c

c

center: ( , )h k ( 2,0)

( 2, 2)

Lesson 10-5

Hyperbolas

Uses of Hyperbolas

http://ccins.camosun.bc.ca/~jbritton/hypplane_lg.GIF

Definition

A hyperbola is the

set of points in a plane

the difference of whose

distances from two fixed

point (foci) is constant.

The point midway between

the two foci is the center.

2d

1d

F FC

Standard Form of a Hyperbola

Standard Form2 2

2 21

x y

a b

2 2

2 21

x h y k

a b

Center: ( , )h k

Vertices: ,h a k

Rectangle:

,

,

h a k

h k b

Focus: ,h c k

2 2 2c a b

F F

Transverse Axis is the x-axis

Standard Form of a Hyperbola

Standard Form2 2

2 21

y x

a b

2 2

2 21

y k x h

a b

Center: ( , )h k

Vertices: ,h k a

Rectangle:

,

,

h k a

h b k

Focus: ,h k c

2 2 2c a b

F

F

Transverse Axis is the y-axis


Graph the equation.2 2

1169 16

y x

Step 1 - Rewrite the equation in standard form.

2 2

2 21

y x

a b

Equation already in standard form


Graph the equation.2 2

1169 16

y x

Step 2 – Identify the a² and b² to determine the

transverse axis.

2 2

2 21

y x

a b

2

2

169

16

a

b

Transverse axis is the y-axis. The graph

is opening up and down.


2 21

y x

a b 2 2

1169 16

y x

Step 3 – Find the center, vertices, rectangle, and focus.

2 169

13

a

a

2 16

4

b

b

Center: , 0,0h k Vertices: , 0,0 13h k a

Rectangle:

, 0,0 13

, 0 4,0

h k a

h b k

Focus:

2 2 2c a b 2 169 16

185 13.60

c

c

, 0,0 185h k c


2 2

2 21

x y

a b

2 225 35 875x y

Graph the equation.

2 225 35 875x y


2 225 35 875

875 875 875

x y

2 2

135 25

x y


2 2

2 21

x y

a b

Step 2 – Identify the a² and b² to determine the

transverse axis.

2

2

35

25

a

b

Transverse axis is the x-axis. The graph

is opening left and right.

2 2

135 25

x y


2 21

x y

a b

2 2

135 25

x y


2 35

35 5.92

a

a

2 25

5

b

b

Center: , 0,0h k Vertices: , 0 35,0h a k

Rectangle:

, 0 35,0

, 0,0 5

h a k

h k b

Focus:

2 2 2c a b 2 35 25

60 7.75

c

c

, 0 60,0h c k

Lesson 10-6

Translating Conic Sections


2 23 6 6 3x x y y

Identify the conic section by writing the equation in

standard form and graph.2 23 6 6 3x x y y Circle or Ellipse


2 23 2 6 3x x y y


2 23 2 6 3x x y y

26

3 92

2 23 2 1 6 9 3 3 9x x y y

22

1 12

2 2

3 1 3 15x y

2 2

3 1 3 15

15 15 15

x y


2 2

3 1 3 15

15 15 15

x y

2 2

1 31

5 15

x y

Ellipse


2 2

1 31

5 15

x y


2

2

15

5

a

b

Center:( , ) ( 1,3)h k Vertices: ( , ) ( 1,3 3.87)h k a

Endpoints:

( , ) ( 1 2.24,3)h b k

Focus:

2 2 2c a b 2 15 5

10 3.16

c

c

( , ) ( 1,3 3.16)h k c

15 3.87

5 2.24

a

b


2 2 14 13x y y


standard form and graph.

2 2 14 13x y y Circle or Ellipse


2 2 14 13x y y


2 2 14 13x y y

22 7 36x y

214

7 492

2 2 14 49 13 49x y y


22 2( )x h y k r

22 7 36x y

Center:

Radius:

( , ) (0, 7)h k

2 36

36 6

r

r


2 24 66y y x x



2 2 6 4 6y x x y Hyperbola


2 214 6 6y y x x


2 214 6 6y y x x

2 214 4 6 9 6 4 9y y x x

24

2 42

2 2

2 3 1y x

26

3 92


2 2

2 3 1y x


2 1a 2 1b

Center: , 3,2h k Vertices: , 3,2 1h k a

Rectangle:

, 3,2 1

, 3 1,2

h k a

h b k

Focus:

2 2 2c a b 2 1 1

2 1.41

c

c

, 3,2 1.41h k c

2 2

2 21

y k x h

a b




2 8 19 0x x y Parabola

2 8 19x x y


2 8 19x x y

28

4 162

2 8 16 19 16x x y


2

4 ( 3)x y


Vertex: , 4,3h k

Focus:

Solve for p

4 1

1

4

p

p

, 4,3 0.25

(4,3.25)

h k p

Directrix:

3 0.25

2.75

y k p

y

y

2( ) 4 ( )x h p y k

2.75y

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Chapter 10wellsmat.startlogic.com/sitebuildercontent/site... · Lesson 10-5 Hyperbolas. Uses of Hyperbolas. Definition A hyperbola is the set of points in a plane the difference of