Worksheet Key 312021 12 34 AM 10 6

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Worksheet Key 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 1

Worksheet Key 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 1

Worksheet Key 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 2

Worksheet Key 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 2

SECTION 10. 6 IDENTIFYING CONIC SECTIONS 3/1/2021 12: 34 AM 10. 6 - Identifying

SECTION 10. 6 IDENTIFYING CONIC SECTIONS 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 3

Recall A. Equations of Conics: B. Put the equations in standard form 3/1/2021 12:

Recall A. Equations of Conics: B. Put the equations in standard form 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 4

Quick Examples Identify whether it is a circle, ellipse, hyperbola, or parabola? 1. 2.

Quick Examples Identify whether it is a circle, ellipse, hyperbola, or parabola? 1. 2. circle ellipse 3. 4. parabola 3/1/2021 12: 34 AM hyperbola 10. 6 - Identifying Conic Sections 5

Steps in Identifying Conics A. B. C. D. 3/1/2021 12: 34 AM Put the

Steps in Identifying Conics A. B. C. D. 3/1/2021 12: 34 AM Put the standard form equation in coefficient ABC order, Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. Identify A, B, and C Apply the discriminant formula, b 2 – 4 ac Determine the answer to identify the proper conic 10. 6 - Identifying Conic Sections 6

Classifying Conic Sections All conic sections can be written in the general form Ax

Classifying Conic Sections All conic sections can be written in the general form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. Circles and Ellipses using b 2 – 4 ac Circles b 2 – 4 ac < 0 b = 0 AND a = c Ellipses b 2 – 4 ac < 0 b ≠ 0 OR a ≠ c 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 7

Classifying Conic Sections All conic sections can be written in the general form Ax

Classifying Conic Sections All conic sections can be written in the general form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. Parabolas and Hyperbolas using b 2 – 4 ac Parabolas b 2 – 4 ac = 0 Hyperbolas 3/1/2021 12: 34 AM b 2 – 4 ac > 0 10. 6 - Identifying Conic Sections 8

Example 1 Identify the conic section that the equation represents and the discriminant, 12

Example 1 Identify the conic section that the equation represents and the discriminant, 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 and show all work A. Put the standard form equation in coefficient ABC order, Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 9

Example 1 Identify the conic section that the equation represents and the discriminant, 12

Example 1 Identify the conic section that the equation represents and the discriminant, 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 and show all work B. Put in A, B, and C 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 A = 12, B = 24, C = 12 3/1/2021 12: 34 AM 10. 6 - Identifying Conic Sections 10

Example 1 Identify the conic section that the equation represents and the discriminant, 12

Example 1 Identify the conic section that the equation represents and the discriminant, 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 and show all work C. Apply the discriminant formula, b 2 – 4 ac A = 12, B = 24, C = 12 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 11

Example 1 Identify the conic section that the equation represents and the discriminant, 12

Example 1 Identify the conic section that the equation represents and the discriminant, 12 x 2 + 24 xy + 12 y 2 + 25 y = 0 and show all work D. Determine the answer to identify the proper conic A = 12, B = 24, Circles b 2 – 4 ac < 0 b = 0 AND a = c Ellipses b 2 – 4 ac < 0 b ≠ 0 OR a ≠ c Parabolas b 2 – 4 ac = 0 Hyperbolas b 2 – 4 ac > 0 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections C = 12 12

Example 2 Identify the conic section that the equation represents and discriminant, 9 x

Example 2 Identify the conic section that the equation represents and discriminant, 9 x 2 + 9 y 2 – 18 x – 12 y – 50 = 0 and show all work 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 13

Example 3 Identify the conic section that the equation represents and the discriminant, 8

Example 3 Identify the conic section that the equation represents and the discriminant, 8 x 2 – 15 xy + 6 y 2 + x – 8 y + 12 = 0 and show all work 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 14

Your Turn Identify the conic section that the equation represents and the discriminant, 4

Your Turn Identify the conic section that the equation represents and the discriminant, 4 x 2 + y 2 = 8 x + 8 and show all work 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 15

Steps in Completing the Square A. B. C. D. E. Identify whether it is

Steps in Completing the Square A. B. C. D. E. Identify whether it is an ellipse by using the equation, b 2 – 4 ac where the answer is less than zero using the equation, 2 2 Ax + Bxy + Cy + Dx + Ey + F = 0 Rearrange variables for x’s and y’s through factoring Take GCF and add everything to other side Use completing the square; using what’s added to the x’s and y’s is added to the radius 1. Identify the coefficient which is raised to the first power and divide the term by 2 2. Take the second term, divide the term by 2, and square that number 3. Add to both sides to the equation 4. Put the equation into factored form Put it in standard form 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 16

Example 4 Change the equation x 2 + y 2 + 8 x –

Example 4 Change the equation x 2 + y 2 + 8 x – 6 y – 11 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 17

Example 4 Change the equation x 2 + y 2 + 8 x –

Example 4 Change the equation x 2 + y 2 + 8 x – 6 y – 11 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 18

Example 4 Change the equation x 2 + y 2 + 8 x –

Example 4 Change the equation x 2 + y 2 + 8 x – 6 y – 11 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 19

Example 5 Change the equation x 2 + 4 y 2 + 10 x

Example 5 Change the equation x 2 + 4 y 2 + 10 x – 8 y + 13 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 20

Example 5 Change the equation x 2 + 4 y 2 + 10 x

Example 5 Change the equation x 2 + 4 y 2 + 10 x – 8 y + 13 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 21

Example 5 Change the equation x 2 + 4 y 2 + 10 x

Example 5 Change the equation x 2 + 4 y 2 + 10 x – 8 y + 13 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 22

Example 5 Change the equation x 2 + 4 y 2 + 10 x

Example 5 Change the equation x 2 + 4 y 2 + 10 x – 8 y + 13 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 23

Example 6 Change the equation 16 x 2 – 9 y 2 – 64

Example 6 Change the equation 16 x 2 – 9 y 2 – 64 x + 18 y – 89 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 24

Example 7 Change the equation x 2 + 14 x – 12 y +

Example 7 Change the equation x 2 + 14 x – 12 y + 97 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 25

Your Turn Change the equation – 4 x 2 + 16 y 2 –

Your Turn Change the equation – 4 x 2 + 16 y 2 – 16 x + 32 y – 64 = 0 to standard form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 26

Example 8 Change the equation (x – 1)2 + (y + 2)2 = 4

Example 8 Change the equation (x – 1)2 + (y + 2)2 = 4 to general form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 27

Example 9 Change the equation and identify the conic 3/1/2021 12: 35 AM 10.

Example 9 Change the equation and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections to general form 28

Example 9 Change the equation and identify the conic 3/1/2021 12: 35 AM 10.

Example 9 Change the equation and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections to general form 29

Example 10 Change the equation 9 x = (y + 8)2 to general form

Example 10 Change the equation 9 x = (y + 8)2 to general form and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 30

Your Turn Change the equation and identify the conic 3/1/2021 12: 35 AM 10.

Your Turn Change the equation and identify the conic 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections to general form 31

Assignment Worksheet 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 32

Assignment Worksheet 3/1/2021 12: 35 AM 10. 6 - Identifying Conic Sections 32

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