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Understanding Conic Sections: Characteristics, Definitions, and Equations

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katoka
31 diapositives · 481 Views · Aug 14, 2014

This summary covers key concepts of conic sections, including parabolas, circles, ellipses, and hyperbolas. It explains how these shapes arise from cutting a cone with a plane and provides the general forms of their equations. Through various examples, it demonstrates how to identify the type of conic section from given equations and how to graph them. The geometric definitions highlight the relationship between the focus, directrix, and eccentricity for each conic type, aiding in a deeper understanding of their properties.

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  1. College Algebra & Trigonometry and Precalculus 4th EDITION

  2. Summary of the Conic Sections 10.4 Characteristics Identifying Conic Sections Geometric Definition of Conic Sections

  3. Characteristics The graphs of parabolas, circles, ellipses, and hyperbolas are called conic sections since each graph can be obtained by cutting a cone with a plane, as suggested by Figure 1 at the beginning of the chapter. All conic sections of the types presented in this chapter have equations of the general form where either A or C must be nonzero.

  4. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 Determine the type of conic section represented by each equation, and graph it. a. Solution Subtract 5y2. Divide each term by 25. Divide by 25.

  5. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 Determine the type of conic section represented by each equation, and graph it. a. Solution The equation represents a hyperbola centered at the origin, with asymptotes Remember both the positive and negative square roots.

  6. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 Determine the type of conic section represented by each equation, and graph it. a. Solution The x-intercepts are  5; the graph is shown here.

  7. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 b. Solution Complete the square on both x and y. Regroup terms. Factor; add 16 and 25.

  8. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 b. Solution The resulting equation is that of a circle with radius 0; that is, the point (4, – 5). If we had obtained a negative number on the right (instead of 0), the equation would have no solution at all, and there would be no graph.

  9. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 c. Solution The coefficients of the x2 -and y2 -terms are unequal and both positive, so the equation might represent an ellipse but not a circle. (It might also represent a single point or no points at all.)

  10. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 c. Solution Factor out 4; factor out 9. Complete the square. Multiply. 4(– 4) = – 16 Distributive property.

  11. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 c. Solution Factor; add 97. Divide by 36. This equation represents an ellipse having center (2, – 3) and graph as shown here.

  12. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 d. Solution Since only one variable is squared (x, andnot y), the equation represents a parabola. Get the term with y (the variable that is not squared) alone on one side.

  13. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 d. Solution Isolate the y-term. Regroup terms: factor out – 1. Complete the square.

  14. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 d. Solution Distributive property; – (– 9) = + 9 Factor; add. Multiply by ⅛. Subtract 2.

  15. DETERMINING TYPES OF CONIC SECTIONS FROM EQUATIONS Example 1 d. Solution The parabola has vertex (3, 2) and opens down, as shown in the graph here. An equivalent form for this parabola is

  16. DETERMINING THE TYPE OF CONIC SECTION FROM ITS EQUATION Example 2 Identify the graph of Solution Factor out 4; factor out – 9. Complete the square.

  17. DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION Example 2 Identify the graph of Solution Distributive property. Factor; add 16 and subtract 9.

  18. DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION Example 2 Identify the graph of Solution Because of the – 36, we might think that this equation does not have a graph. However, the minus sign in the middle on the left shows that the graph is that of a hyperbola.

  19. DETERMINING THE TYPEOF CONIC SECTION FROM ITS EQUATION Example 2 Identify the graph of Be careful here. Solution Divide by – 36; rearrange terms. This hyperbola has center (1, 2). The graph is shown here.

  20. Geometric Definition of Conic Sections In Section 6.1, aparabola was defined as the set of points in a plane equidistant from a fixed point (focus) and a fixed line (directrix). A parabola has eccentricity 1. This definition can be generalized to apply to the ellipse and the hyperbola.

  21. Geometric Definition of Conic Sections This figure shows an ellipse with a = 4, c = 2, and e = ½. The line x = 8 is shown also. For any point P on the ellipse,

  22. Geometric Definition of Conic Sections This figure shows a hyperbola with a = 2, c = 4, and e = 2, along with the line x = 1. For any point P on the hyperbola. The following geometric definition applies to all conic sections except circles, which have e = 0.

  23. Geometric Definition of a Conic Section Given a fixed point F (focus), a fixed line L (directrix), and a positive number e, the set of all points P in the plane such that is a conic section of eccentricity e. The conic section is a parabola when e = 1, an ellipse when 0 < e < 1, and a hyperbola when e > 1.