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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
Uses of Parabolas
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
Example 5 – Page 547, #24
21
24x y
Step 2 – Rewrite the equation in standard form.
2 4y px
2
1 24
x y
2 24y x
Example 5 – Page 547, #24
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
Example 5 – Page 547, #30
Identify the vertex, the focus, and the directrix and sketch
the graph.
2( 2) 4x y
Step 1 – Locate which variable has the square.
-term the parabola has vertical symmetry (y-axis)2x
Example 5 – Page 547, #30
2( 2) 4x y
Step 2 – Rewrite the equation in standard form.
2( ) 4 ( )x h p y k
Already in standard form.
Example 5 – Page 547, #30
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
Example 5 – Page 547, #34
Identify the vertex, the focus, and the directrix and sketch
the graph.2 24 8 16x y x
Step 1 – Locate which variable has the square.
-term the parabola has vertical symmetry (y-axis)2x
Example 5 – Page 547, #34
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
Example 5 – Page 547, #34
2 4( ) ( )px h y k 2( 4) 24x y
Step 3 – Identify the vertex, focus and directrix.
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
Example 5, Page 553, #30
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
Example 5, Page 553, #62
2 22 1 3 1x x y
Find the center and the radius of the circle.
2 22 1 4x x y
Step 1 – Rewrite the equation in standard form.
2 22 3x x y
2 2 32x x y
221 1
2
22 2( )x h y k r
Example 5, Page 553, #62
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
Uses of Ellipses
Uses of Ellipses
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
Example 3, Page 559, #18
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
Example 3, Page 559, #18
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
Example 3, Page 559, #24
Find the foci and graph the ellipse
2 24 16x y
Step 1 – Rewrite the equation in standard form
2 24 16
16 16 16
x y
2 2
116 4
x y
2 2
2 21
x y
a b
Example 3, Page 559, #24
Step 2 – Identify a² and b² to determine which axis is the
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
Example 3, Page 559, #242 2
116 4
x y
2
2
16
4
a
b
Step 3 – Find the center, vertices, endpoints and focus.
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
Example 3, Page 559, #40
Find the foci of the ellipse
2 22 8 4 0x x y
Step 1 – Rewrite the equation in standard form
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
Example 3, Page 559, #40
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
Example 3, Page 559, #40
2 22
12 4
x y
Step 2 – Identify a² and b² to determine which axis is the
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
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
Example 1, Page 566, #2
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
Example 1, Page 566, #2
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.
Example 1, Page 566, #22 2
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
Example 1, Page 566, #8
2 2
2 21
x y
a b
2 225 35 875x y
Graph the equation.
2 225 35 875x y
Step 1 - Rewrite the equation in standard form.
2 225 35 875
875 875 875
x y
2 2
135 25
x y
Example 1, Page 566, #8
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
Example 1, Page 566, #82 2
2 21
x y
a b
2 2
135 25
x y
Step 3 – Find the center, vertices, rectangle, and focus.
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
Example 4, Page 574, #14
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
Step 1 - Rewrite the equation in standard form.
2 23 2 6 3x x y y
Example 4, Page 574, #14
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
Example 4, Page 574, #14
2 2
3 1 3 15
15 15 15
x y
2 2
1 31
5 15
x y
Ellipse
Example 4, Page 574, #14
2 2
1 31
5 15
x y
Step 2 – Find the center, vertices, endpoints and focus.
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
Example 4, Page 574, #18
2 2 14 13x y y
Identify the conic section by writing the equation in
standard form and graph.
2 2 14 13x y y Circle or Ellipse
Step 1 - Rewrite the equation in standard form.
2 2 14 13x y y
Example 4, Page 574, #18
2 2 14 13x y y
22 7 36x y
214
7 492
2 2 14 49 13 49x y y
Example 4, Page 574, #18
22 2( )x h y k r
22 7 36x y
Center:
Radius:
( , ) (0, 7)h k
2 36
36 6
r
r
Example 4, Page 574, #16
2 24 66y y x x
Identify the conic section by writing the equation in
standard form and graph.
2 2 6 4 6y x x y Hyperbola
Step 1 - Rewrite the equation in standard form.
2 214 6 6y y x x
Example 4, Page 574, #16
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
Example 4, Page 574, #16
2 2
2 3 1y x
Step 2 – Find the center, vertices, rectangle, and focus.
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
Example 4, Page 574, #12
Identify the conic section by writing the equation in
standard form and graph.
2 8 19 0x x y Parabola
2 8 19x x y
Step 1 - Rewrite the equation in standard form.
2 8 19x x y
28
4 162
2 8 16 19 16x x y
Example 4 – Page 574, #12
2
4 ( 3)x y
Step 2 – Identify the vertex, focus and directrix.
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