Conic Review: Circles

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Conic Review Circles
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Conic Review Ellipses (2 slides)
If a gt b
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The standard form of an ellipse centered at (0,
0) depends on whether the major axis is
horizontal or vertical.
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Conic Review Hyperbolas (2 slides)
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The standard form of the equation of a hyperbola
depends on whether the hyperbolas transverse
axis is horizontal or vertical.
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In Lesson 10-2 through 10-5, you learned about
the four conic sections. Recall the equations of
conic sections in standard form. In these forms,
the characteristics of the conic sections can be
identified.
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Example 1 Identifying Conic Sections in Standard
Form
Identify the conic section that each equation
represents.
A.
This equation is of the same form as a parabola
with a horizontal axis of symmetry.
B.
This equation is of the same form as a hyperbola
with a horizontal transverse axis.
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Example 1 Identifying Conic Sections in Standard
Form
Identify the conic section that each equation
represents.
C.
This equation is of the same form as a circle.
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Example 2A Finding the Standard Form of the
Equation for a Conic Section
Find the standard form of the equation by
completing the square. Then identify and graph
each conic.
x2 y2 8x 10y 8 0
Rearrange to prepare for completing the square in
x and y.
Complete both squares.
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Example 2A Continued
(x 4)2 (y 5)2 49
Factor and simplify.
Because the conic is of the form (x h)2 (y
k)2 r2, it is a circle with center (4, 5) and
radius 7.
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Example 2B Finding the Standard Form of the
Equation for a Conic Section
Find the standard form of the equation by
completing the square. Then identify and graph
each conic.
5x2 20y2 30x 40y 15 0
Rearrange to prepare for completing the square in
x and y.
Factor 5 from the x terms, and factor 20 from the
y terms.
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Example 2B Continued
Complete both squares.
5(x 3)2 20(y 1)2 80
Factor and simplify.
Divide both sides by 80.
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Example 2B Continued
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Example 3 Aviation Application
An airplane makes a dive that can be modeled by
the equation 9x2 25y2 18x 50y 209 0
with dimensions in hundreds of feet. How close to
the ground does the airplane pass?
The graph of 9x2 25y2 18x 50y 209 0 is a
conic section. Write the equation in standard
form.
Rearrange to prepare for completing the square in
x and y.
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Example 3 Continued
Factor 9 from the x terms, and factor 25 from
the y terms.
Complete both squares.
25(y 1)2 9(x 1)2 225
Simplify.
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Example 3 Continued
Divide both sides by 225.
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Notes
Identify the conic section that each equation
represents.
1. x2 (y 14)2 112
This equation is of the same form as a circle.
2.
This equation is of the same form as a hyperbola
with a vertical transverse axis.
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Notes
3. Find the standard form of 4x2 y2 8x 8y
16 0 by completing the square. Then identify
and graph the conic.
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Notes
4. Find the standard form of the equation by
completing the square. Then identify and graph
y2 9x 16y 64 0
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Conic Review Parabolas (2 slides)
A parabola is the set of all points P(x, y) in a
plane that are an equal distance from both a
fixed point, the focus, and a fixed line, the
directrix. A parabola has a axis of symmetry
perpendicular to its directrix and that passes
through its vertex. The vertex of a parabola is
the midpoint of the perpendicular segment
connecting the focus and the directrix.
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In Lesson 10-2 through 10-5, you learned about
the four conic sections. Recall the equations of
conic sections in standard form. In these forms,
the characteristics of the conic sections can be
identified.
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Conic Review Extra Info
The following power-point slides contain extra
examples and information.
Review of Lesson Objectives Identify and
transform conic functions. Use the method of
completing the square to identify and graph conic
sections.
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Check It Out! Example 3a Continued
(y 8)2 9x
Factor and simplify.
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Check It Out! Example 3b
Find the standard form of the equation by
completing the square. Then identify and graph
each conic.
16x2 9y2 128x 108y 436 0
Rearrange to prepare for completing the square in
x and y.
Factor 16 from the x terms, and factor 9 from the
y terms.
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Check It Out! Example 3b Continued
Complete both squares.
16(x 4)2 9(y 6)2 144
Factor and simplify.
Divide both sides by 144.
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Check It Out! Example 3b Continued
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Check It Out! Example 4
An airplane makes a dive that can be modeled by
the equation 16x2 9y2 96x 36y 252 0,
measured in hundreds of feet. How close to the
ground does the airplane pass?
The graph of 16x2 9y2 96x 36y 252 0 is a
conic section. Write the equation in standard
form.
Rearrange to prepare for completing the square in
x and y.
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Check It Out! Example 4 Continued
Factor 16 from the x terms, and factor 9 from
the y terms.
Complete both squares.
16(x 3)2 9(y 2)2 144
Simplify.
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Check It Out! Example 4 Continued
Divide both sides by 144.
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All conic sections can be written in the general
form Ax2 Bxy Cy2 Dx Ey F 0. The conic
section represented by an equation in general
form can be determined by the coefficients.
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Example 2A Identifying Conic Sections in General
Form
Identify the conic section that the equation
represents.
4x2 10xy 5y2 12x 20y 0
Identify the values for A, B, and C.
A 4, B 10, C 5
B2 4AC
Substitute into B2 4AC.
(10)2 4(4)(5)
Simplify.
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Because B2 4AC gt 0, the equation represents a
hyperbola.
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Example 2B Identifying Conic Sections in General
Form
Identify the conic section that the equation
represents.
9x2 12xy 4y2 6x 8y 0.
Identify the values for A, B, and C.
A 9, B 12, C 4
B2 4AC
Substitute into B2 4AC.
(12)2 4(9)(4)
Simplify.
0
Because B2 4AC 0, the equation represents a
parabola.
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Example 2C Identifying Conic Sections in General
Form
Identify the conic section that the equation
represents.
8x2 15xy 6y2 x 8y 12 0
Identify the values for A, B, and C.
A 8, B 15, C 6
B2 4AC
(15)2 4(8)(6)
Substitute into B2 4AC.
Simplify.
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Because B2 4AC gt 0, the equation represents a
hyperbola.
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Check It Out! Example 2a
Identify the conic section that the equation
represents.
9x2 9y2 18x 12y 50 0
Identify the values for A, B, and C.
A 9, B 0, C 9
Substitute into B2 4AC.
B2 4AC
(0)2 4(9)(9)
Simplify. The conic is either a circle or an
ellipse.
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A C
Because B2 4AC lt 0 and A C, the equation
represents a circle.
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Check It Out! Example 2b
Identify the conic section that the equation
represents.
12x2 24xy 12y2 25y 0
Identify the values for A, B, and C.
A 12, B 24, C 12
B2 4AC
Substitute into B2 4AC.
242 4(12)(12)
Simplify.
0
Because B2 4AC 0, the equation represents a
parabola.