UNIT 13.6 CONICSUNIT 13.6 CONICSFROM EQUATIONSFROM EQUATIONS

Warm UpSolve by completing the square.

1. x2 + 6x = 91

2. 2x2 + 8x – 90 = 0

Identify and transform conic functions.

Use the method of completing the square to identify and graph conic sections.

Objectives

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.

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.

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.

Identify the conic section that each equation represents.

Check It Out! Example 1

a. x2 + (y + 14)2 = 112

– = 1 (y – 6)2

22

(x – 1)2

212b.

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.

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.

4x2 – 10xy + 5y2 + 12x + 20y = 0

A = 4, B = –10, C = 5

B2 – 4AC

Substitute into B2 – 4AC.(–10)2 – 4(4)(5)

Simplify.

Because B2 – 4AC > 0, the equation represents a hyperbola.

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.

9x2 – 12xy + 4y2 + 6x – 8y = 0.

A = 9, B = –12, C = 4

B2 – 4AC

Substitute into B2 – 4AC.(–12)2 – 4(9)(4)

Simplify.

Because B2 – 4AC = 0, the equation represents a parabola.

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.

8x2 – 15xy + 6y2 + x – 8y + 12 = 0

A = 8, B = –15, C = 6

B2 – 4AC

Substitute into B2 – 4AC.(–15)2 – 4(8)(6)

Simplify.

Because B2 – 4AC > 0, the equation represents a hyperbola.

Identify the conic section that the equation represents.

9x2 + 9y2 – 18x – 12y – 50 = 0

Check It Out! Example 2a

Identify the conic section that the equation represents.

12x2 + 24xy + 12y2 + 25y = 0

Check It Out! Example 2b

You must factor out the leading coefficient of x2 and y2 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.

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.

x2 + y2 + 8x – 10y – 8 = 0

x2 + 8x + + y2 – 10y + = 8 + +

Complete both squares.2

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.

Example 3B: Finding the Standard Form of the Equation for a Conic Section

Rearrange to prepare for completing the square in x and y.

5x2 + 20y2 + 30x + 40y – 15 = 0

5x2 + 30x + + 20y2 + 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.

Example 3B Continued

Complete both squares.

5(x + 3)2 + 20(y + 1)2 = 80 Factor and simplify.

Divide both sides by 80.

65 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

Because the conic is of the form (x – h)2

a2+ = 1,(y – k)2

b2

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

Find the standard form of the equation by completing the square. Then identify and graph each conic.

y2 – 9x + 16y + 64 = 0

Check It Out! Example 3a

Check It Out! Example 3a Continued

16x2 + 9y2 – 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.

Check It Out! Example 3b Continued

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