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Identifying Conic Sections

Identifying Conic Sections. Section 10-6. 1. 8/22/2014 5:52 AM. 10.6 - Identifying Conic Sections. Recall. Equations of Conics: Put the equations in standard form. Quick Examples. Identify whether it is a circle, ellipse, hyperbola, or parabola? 1. 2. 3. 4. c i r c l e.

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Identifying Conic Sections

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  1. Identifying Conic Sections Section 10-6 1 8/22/2014 5:52 AM 10.6 - Identifying Conic Sections

  2. Recall • Equations of Conics: • Put the equations in standard form 10.6 - Identifying Conic Sections

  3. Quick Examples Identify whether it is a circle, ellipse, hyperbola, or parabola? 1. 2. 3. 4. c i r c l e e l l i p s e p a r a b o l a h y p e r b o l a 10.6 - Identifying Conic Sections

  4. Steps to Identifying Conics • Put the standard form equation in coefficient ABC order, Ax2 + Bxy + Cy2 + Dx + Ey+ F = 0. • Identify A, B, and C • Apply the discriminant formula, b2 – 4ac • Determine the answer to identify the proper conic 10.6 - Identifying Conic Sections

  5. Classifying Conic Sections All conic sections can be written in the general form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. 10.6 - Identifying Conic Sections

  6. Classifying Conic Sections All conic sections can be written in the general form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. 10.6 - Identifying Conic Sections

  7. Example 1 Identify the conic section that the equation represents and the discriminant, 12x2 + 24xy + 12y2 + 25y = 0 Put the standard form equation in coefficient ABC order, Ax2 + Bxy + Cy2 + Dx + Ey+ F = 0. 12x2 + 24xy + 12y2 + 25y = 0 10.6 - Identifying Conic Sections

  8. Example 1 Identify the conic section that the equation represents and the discriminant, 12x2 + 24xy + 12y2 + 25y = 0 2. IdentifyA, B, and C 12x2 + 24xy + 12y2 + 25y = 0 12x2 + 24xy + 12y2 + 25y = 0 A = 12, B = 24, C = 12 10.6 - Identifying Conic Sections

  9. Example 1 Identify the conic section that the equation represents and the discriminant, 12x2 + 24xy + 12y2 + 25y = 0 3. Apply the discriminant formula, b2 – 4ac A = 12, B = 24, C = 12 10.6 - Identifying Conic Sections

  10. Example 1 Identify the conic section that the equation represents and the discriminant, 12x2 + 24xy + 12y2 + 25y = 0 4. Determine the answer to identify the proper conic A = 12, B = 24, C = 12 10.6 - Identifying Conic Sections

  11. Example 1 Identify the conic section that the equation represents and the discriminant, 12x2 + 24xy + 12y2 + 25y = 0 10.6 - Identifying Conic Sections

  12. Example 2 Identify the conic section that the equation represents, 9x2 + 9y2 – 18x – 12y – 50 = 0 10.6 - Identifying Conic Sections

  13. Your Turn Identify the conic section that the equation represents and the discriminant, 8x2– 15xy + 6y2 + x – 8y + 12 = 0 10.6 - Identifying Conic Sections

  14. Example 3 Identify the conic section that the equation represents and the discriminant, 4x2 + y2 = 8x + 8 10.6 - Identifying Conic Sections

  15. Steps for Completing the Square • Identify the conic in standard form using the discriminant formula • 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 • Put it in standard form 10.6 - Identifying Conic Sections

  16. Example 4 Change the equation to standard form and identify the conic 10.6 - Identifying Conic Sections

  17. Example 4 Change the equation to standard form and identify the conic 10.6 - Identifying Conic Sections

  18. Example 4 Change the equation to standard form and identify the conic How does one get the factors? We use COMPLETING THE SQUARE. Remember we take the second term and divide it by 2. Then we square it. 10.6 - Identifying Conic Sections

  19. Example 4 Change the equation to standard form and identify the conic How does one get the factors? We use COMPLETING THE SQUARE. Remember we take the second term and divide it by 2. Then we square it. 10.6 - Identifying Conic Sections

  20. 5x2 + 30x + + 20y2 + 40y + = 15 + + 5(x2 + 6x + )+ 20(y2 + 2y + ) = 15 + + Example 5 Change the equation 5x2 + 20y2 + 30x + 40y – 15 = 0 to standard form Use Discriminant Formula to identify the conic a = 5, b = 0, c =20; b2 – 4ac (0)2 – 4(5)(20); -400 < 0; a ≠ c; ELLIPSE Rearrange to prepare for completing the square in x and y. Factor 5 from the x terms, and factor 20 from the y terms. 10.6 - Identifying Conic Sections

  21. ( ) ( ) x + 3 2 y + 1 2 + = 1 16 4 Example 5 Change the equation 5x2 + 20y2 + 30x + 40y – 15 = 0 to standard form 5(x +3)2 + 20(y + 1)2 = 80 Factor and simplify. Divide both sides by 80. 10.6 - Identifying Conic Sections

  22. y2 + 16y + = 9x – 64 + Add , or 64, to both sides to complete the square. Example 6 Change the equation y2 – 9x + 16y + 64 = 0 to standard form Rearrange to prepare for completing the square in y. 10.6 - Identifying Conic Sections

  23. Example 6 Change the equation y2 – 9x + 16y + 64 = 0 to standard form (y +8)2 = 9x 10.6 - Identifying Conic Sections

  24. Your Turn Change the equation x2 + 4y2 – 6x – 7 = 0 to standard form and identify the conic 10.6 - Identifying Conic Sections

  25. Practice Problems Convert the following problems to standard form: • 6x2 – 5y2 + 24x + 20y – 56 = 0 3. 4x2 + 48x + y + 156 = 0 10.6 - Identifying Conic Sections

  26. Assignment 26 Page 764 15-31 odd, 35, 43 8/22/2014 5:52 AM 10.6 - Identifying Conic Sections

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