Identifying Conic Sections

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10-6
Identifying Conic Sections
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Solve by completing the square.
1. x2 6x 91
2. 2x2 8x 90 0
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Objectives
Identify and transform conic functions. Use the
method of completing the square to identify and
graph conic sections.
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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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Check It Out! Example 1
Identify the conic section that each equation
represents.
a. x2 (y 14)2 112
b.
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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.
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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
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Check It Out! Example 2b
Identify the conic section that the equation
represents.
12x2 24xy 12y2 25y 0
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If you are given the equation of a conic in
standard form, you can write the equation in
general form by expanding the binomials. If you
are given the general form of a conic section,
you can use the method of completing the square
from Lesson 5-4 to write the equation in standard
form.
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Example 3A 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 3A 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 3B 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 3B 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 3B Continued
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Check It Out! Example 3a
Find the standard form of the equation by
completing the square. Then identify and graph
each conic.
y2 9x 16y 64 0
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Check It Out! Example 3a Continued
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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
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Check It Out! Example 3b Continued