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Understanding Conic Sections: General Second-Degree Equations and Discriminant Analysis

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jaime-mcdowell
19 diapositives · 425 Views · Nov 17, 2014

This guide explores the general second-degree equation for conic sections, represented as Ax² + Bxy + Cy² + Dx + Ey + F = 0. It explains how to classify conics—circles, ellipses, parabolas, and hyperbolas—using the discriminant, B² - 4AC. Learn the conditions for different conics based on the coefficients of the equation and how to complete the square to rewrite equations in standard form. Understanding these concepts will enhance your ability to recognize and analyze conic sections effectively.

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Understanding Conic Sections: General Second-Degree Equations and Discriminant Analysis
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  1. Classifying Conics10.6 What is the general 2nd degree equation for any conic? What information can the discriminant tell you about a conic?

  2. The equation of any conic can be written in the form- Called a general 2nd degree equation

  3. Circles Can be multiplied out to look like this….

  4. Ellipse Can be written like this…..

  5. Parabola Can be written like this…..

  6. Hyperbola Can be written like this…..

  7. How do you know which conic it is when it’s been multiplied out? • Pay close attention to whose squared and whose not… • Look at the coefficients in front of the squared terms and their signs.

  8. Circle Both x and y are squared And their coefficients are the same number and sign

  9. Ellipse • Both x and y are squared • Their coefficients are different but their signs remain the same.

  10. Parabola • Either x or y is squared but not both

  11. Hyperbola • Both x and y are squared • Their coefficients are different and so are their signs.

  12. You Try! • Ellipse • Parabola • Hyperbola • Circle • Hyperbola • Parabola • Circle • Ellipse • Hyperbola • Ellipse

  13. When you want to be sure… of a conic equation, then find the type of conic using discriminate information: Ax2 +Bxy +Cy2 +Dx +Ey +F = 0 B2 − 4AC < 0, B = 0 & A = C Circle B2 − 4AC < 0 & either B≠0 or A≠C Ellipse B2 − 4AC = 0 Parabola B2 − 4AC > 0 Hyperbola

  14. Classify the Conic 2x2 + y2−4x − 4 = 0 Ax2 +Bxy +Cy2 +Dx +Ey +F = 0 A = 2 B = 0 C = 1 B2 − 4AC = 02 − 4(2)(1) = −8 B2 − 4AC < 0, the conic is an ellipse

  15. Graph the Conic 2x2 + y2−4x − 4 = 0 2x2−4x + y2 = 4 2(x2−2x +___)+ y2 = 4 + ___ (−2/2)2= 1 2(x2−2x +1)+ y2 = 4 + 2(1) 2(x−1)2 + y2 = 6 V(1±√6), CV(1±√3) Complete the Square

  16. Steps to Complete the Square 1. Group x’s and y’s. (Boys with the boys and girls with the girls) Send constant numbers to the other side of the equal sign. 2. The coefficient of the x2 and y2 must be 1. If not, factor out. 3. Take the number before the x, divide by 2 and square. Do the same with the number before y. 4. Add these numbers to both sides of the equation. *(Multiply it by the common factor in #2) 5. Factor

  17. Write the equation in standard form by completing the square

  18. What is the general 2nd degree equation for any conic? What information can the discriminant tell you about a conic? B2- 4AC < 0, B = 0, A = C Circle B2- 4AC < 0, B ≠ 0, A ≠ C Ellipse B2- 4AC = 0, Parabola B2- 4AC > 0 Hyperbola

  19. Assignment 10.6 Page 628, 29-55 odd