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10-6
10-6Identifying
IdentifyingConic
ConicSections
Sections
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
10-6 Identifying Conic Sections
Warm Up
Solve by completing the square.
1. x2 + 6x = 91
2. 2x2 + 8x – 90 = 0
Holt Algebra 2
10-6 Identifying Conic Sections
Objectives
Identify and transform conic functions.
Use the method of completing the
square to identify and graph conic
sections.
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
Holt Algebra 2
10-6 Identifying Conic Sections
Example 1: Identifying Conic Sections in Standard
Form
Identify the conic section that each equation
represents.
A.
(y – 2)2
x+4 =
10
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.
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 1
Identify the conic section that each equation
represents.
a.
x2 + (y + 14)2 = 112
2
(y – 6)2
(x
–
1)
b.
–
= 1
22
212
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
Example 2A: Identifying Conic Sections in General
Form
Identify the conic section that the equation
represents.
4x2 – 10xy + 5y2 + 12x + 20y = 0
A = 4, B = –10, C = 5 Identify the values for A, B, and C.
B2 – 4AC
2
(–10) – 4(4)(5)
Substitute into B2 – 4AC.
20
Simplify.
Because B2 – 4AC > 0, the equation represents a
hyperbola.
Holt Algebra 2
10-6 Identifying Conic Sections
Example 2B: Identifying Conic Sections in General
Form
Identify the conic section that the equation
represents.
9x2 – 12xy + 4y2 + 6x – 8y = 0.
A = 9, B = –12, C = 4 Identify the values for A, B, and C.
B2 – 4AC
2
(–12) – 4(9)(4)
Substitute into B2 – 4AC.
0
Simplify.
Because B2 – 4AC = 0, the equation represents a
parabola.
Holt Algebra 2
10-6 Identifying Conic Sections
Example 2C: Identifying Conic Sections in General
Form
Identify the conic section that the equation
represents.
8x2 – 15xy + 6y2 + x – 8y + 12 = 0
A = 8, B = –15, C = 6
Identify the values for A, B, and C.
B2 – 4AC
(–15)2 – 4(8)(6)
Substitute into B2 – 4AC.
33
Simplify.
Because B2 – 4AC > 0, the equation represents a
hyperbola.
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 2a
Identify the conic section that the equation
represents.
2
2
9x + 9y – 18x – 12y – 50 = 0
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 2b
Identify the conic section that the equation
represents.
12x2 + 24xy + 12y2 + 25y = 0
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Remember!
You must factor out the leading coefficient of x2
and y2 before completing the square.
Holt Algebra 2
10-6 Identifying Conic Sections
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.
x2 + 8x +
+ y2 – 10y +
Complete both squares.
2
Holt Algebra 2
=8+
+
10-6 Identifying Conic Sections
Example 3A Continued
2
2
(x + 4) + (y – 5) = 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.
Holt Algebra 2
10-6 Identifying Conic Sections
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.
5x2 + 30x +
+ 20y2 + 40y +
= 15 +
+
Factor 5 from the x terms, and factor 20 from the y terms.
5(x2 + 6x +
Holt Algebra 2
)+ 20(y2 + 2y +
) = 15 +
+
10-6 Identifying Conic Sections
Example 3B Continued
Complete both squares.
2
2



6
2


2
2
5  x + 6x +    + 20  y + 2y +  
 2  


 2
5(x + 3)2 + 20(y + 1)2 = 80
(x + 3)2 + (y +1 )2 =
16
Holt Algebra 2
4
1
2

6
 
 2
 = 15 + 5   + 20  
2

 2
Factor and simplify.
Divide both sides by
80.
10-6 Identifying Conic Sections
Example 3B Continued
(x – h)2
(y – k)2
+
= 1,
Because the conic is of the form
2
2
a
b
it is an ellipse with center (–3, –1), horizontal major
axis length 8, and minor axis length 4. The covertices are (–3, –3) and (–3, 1), and the vertices
are (–7, –1) and (1, –1).
Holt Algebra 2
10-6 Identifying Conic Sections
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
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 3a Continued
Holt Algebra 2
10-6 Identifying Conic Sections
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
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 3b Continued
Holt Algebra 2