This document discusses identifying conic sections from their equations. It explains how to write equations of conic sections in standard and general form and use the method of completing the square to transform between the forms. Examples show how to identify equations as representing parabolas, circles, ellipses or hyperbolas based on their coefficients and whether they satisfy certain conditions. The objectives are to identify conic sections from their equations in standard or general form and graph the conic sections.

1. UNIT 13.6 CONICSUNIT 13.6 CONICS
FROM EQUATIONSFROM EQUATIONS

2. Warm Up
Solve by completing the square.
1. x
2
+ 6x = 91
2. 2x
2
+ 8x – 90 = 0

3. Identify and transform conic functions.
Use the method of completing the
square to identify and graph conic
sections.
Objectives

4. 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.

6. Identify the conic section that each equation
represents.
Example 1: Identifying Conic Sections in Standard
Form
A.
This equation is of the same form as a parabola
with a horizontal axis of symmetry.
x + 4 = (y – 2)
2
10
B.
This equation is of the same form as a hyperbola
with a horizontal transverse axis.

7. Identify the conic section that each equation
represents.
Example 1: Identifying Conic Sections in Standard
Form
This equation is of the same form as a circle.
C.

8. Identify the conic section that each equation
represents.
Check It Out! Example 1
a. x
2
+ (y + 14)
2
= 11
2
– = 1
(y – 6)
2
2
2
(x – 1)
2
21
2
b.

9. All conic sections can be written in the general form
Ax
2
+ Bxy + Cy
2
+ Dx + Ey+ F = 0. The conic section
represented by an equation in general form can be
determined by the coefficients.

10. Identify the conic section that the equation
represents.
Example 2A: Identifying Conic Sections in General
Form
20
Identify the values for A, B, and C.
4x
2
– 10xy + 5y
2
+ 12x + 20y = 0
A = 4, B = –10, C = 5
B
2
– 4AC
Substitute into B
2
– 4AC.(–10)
2
– 4(4)(5)
Simplify.
Because B
2
– 4AC > 0, the equation represents a
hyperbola.

11. Identify the conic section that the equation
represents.
Example 2B: Identifying Conic Sections in General
Form
0
Identify the values for A, B, and C.
9x
2
– 12xy + 4y
2
+ 6x – 8y = 0.
A = 9, B = –12, C = 4
B
2
– 4AC
Substitute into B
2
– 4AC.(–12)
2
– 4(9)(4)
Simplify.
Because B
2
– 4AC = 0, the equation represents a
parabola.

12. Identify the conic section that the equation
represents.
Example 2C: Identifying Conic Sections in General
Form
33
Identify the values for A, B, and C.
8x
2
– 15xy + 6y
2
+ x – 8y + 12 = 0
A = 8, B = –15, C = 6
B
2
– 4AC
Substitute into B
2
– 4AC.(–15)
2
– 4(8)(6)
Simplify.
Because B
2
– 4AC > 0, the equation represents a
hyperbola.

13. Identify the conic section that the equation
represents.
9x
2
+ 9y
2
– 18x – 12y – 50 = 0
Check It Out! Example 2a

14. Identify the conic section that the equation
represents.
12x
2
+ 24xy + 12y
2
+ 25y = 0
Check It Out! Example 2b

15. You must factor out the leading coefficient of x
2
and y
2
before completing the square.
Remember!
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.

16. Find the standard form of the equation by
completing the square. Then identify and
graph each conic.
Example 3A: Finding the Standard Form of the
Equation for a Conic Section
Rearrange to prepare for completing the square in x and y.
x
2
+ y
2
+ 8x – 10y – 8 = 0
x
2
+ 8x + + y
2
– 10y + = 8 + +
Complete both squares.
2

17. 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
= r
2
,
it is a circle with center (–4, 5) and radius 7.

18. Example 3B: Finding the Standard Form of the
Equation for a Conic Section
Rearrange to prepare for completing the square in x and y.
5x
2
+ 20y
2
+ 30x + 40y – 15 = 0
5x
2
+ 30x + + 20y
2
+ 40y + = 15 + +
Factor 5 from the x terms, and factor 20 from the y terms.
5(x2
+ 6x + )+ 20(y2
+ 2y + ) = 15 + +
Find the standard form of the equation by
completing the square. Then identify and
graph each conic.

19. Example 3B Continued
Complete both squares.
5(x + 3)
2
+ 20(y + 1)
2
= 80 Factor and simplify.
Divide both sides by
80.
6
5 x2
+ 6x + + 20 y2
+ 2y + = 15 + 5 + 20
2
2
2
2
 















6
2









2
2






2 2



( ) ( )
1
16 4
x + 3 2
y +1 2
+ =

20. Because the conic is of the form (x – h)
2
a
2
+ = 1,(y – k)
2
b
2
it is an ellipse with center (–3, –1), horizontal major
axis length 8, and minor axis length 4. The co-
vertices are (–3, –3) and (–3, 1), and the vertices
are (–7, –1) and (1, –1).
Example 3B Continued

21. Find the standard form of the equation by
completing the square. Then identify and
graph each conic.
y
2
– 9x + 16y + 64 = 0
Check It Out! Example 3a

22. Check It Out! Example 3a Continued

23. 16x
2
+ 9y
2
– 128x + 108y + 436 = 0
Check It Out! Example 3b
Find the standard form of the equation by
completing the square. Then identify and
graph each conic.

24. Check It Out! Example 3b Continued

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