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Chapter 9

Chapter 9. Analytic Geometry. Section 9-1. Distance and Midpoint Formulas. Pythagorean Theorem. If the length of the hypotenuse of a right triangle is c, and the lengths of the other two sides are a and b , then c 2 = a 2 + b 2. Example.

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Chapter 9

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  1. Chapter 9 Analytic Geometry

  2. Section 9-1 Distance and Midpoint Formulas

  3. Pythagorean Theorem • If the length of the hypotenuse of a right triangle is c, and the lengths of the other two sides are a and b, then c2 = a2 + b2

  4. Example Find the distance between point D and point F.

  5. Distance Formula D = √(x2 – x1)2 + (y2 – y1)2

  6. Example • Find the distance between points A(4, -2) and B(7, 2) • d = 5

  7. Midpoint Formula M( x1 + x2, y1 + y2) 2 2

  8. Example • Find the midpoint of the segment joining the points (4, -6) and (-3, 2) • M(1/2, -2)

  9. Section 9-2 Circles

  10. Conics • Are obtained by slicing a double cone • Circles, Ellipses, Parabolas, and Hyperbolas

  11. Equation of a Circle The circle with center (h,k) and radius r has the equation (x – h)2 + (y – k)2 = r2

  12. Example • Find an equation of the circle with center (-2,5) and radius 3. • (x + 2)2 + (y – 5)2 = 9

  13. Translation • Sliding a graph to a new position in the coordinate plane without changing its shape

  14. Translation

  15. Example Graph (x – 2)2 + (y + 6)2 = 4

  16. Example • If the graph of the equation is a circle, find its center and radius. • x2 + y2 + 10x – 4y + 21 = 0

  17. Section 9-3 Parabolas

  18. Parabola • A set of all points equidistant from a fixed line called the directrix, and a fixed point not on the line, called the focus

  19. Vertex • The midpoint between the focus and the directrix.

  20. Parabola - Equations y-k = a(x-h)2 Vertex (h,k) symmetry x = h x - h = a(y-k)2 Vertex (h,k) symmetry y = k

  21. Equation of a Parabola • Remember: y – k = a(x – h)2 • (h,k) is the vertex of the parabola

  22. Example 1 • The vertex of a parabola is (-5, 1) and the directrix is the line y = -2. Find the focus of the parabola. • (-5 4)

  23. Example 1

  24. Example 2 • Find an equation of the parabola having the point F(0, -2) as the focus and the line x = 3 as the directrix.

  25. y – k = a(x – h)2 • a = 1/4c where c is the distance between the vertex and focus • Parabola opens upward if a>0, and downward if a< 0

  26. y – k = a(x – h)2 • Vertex (h, k) • Focus (h, k+c) • Directrix y = k – c • Axis of Symmetry x = h

  27. x - h = a(y –k)2 • a = 1/4c where c is the distance between the vertex and focus • Parabola opens to the right if a>0, and to the left if a< 0

  28. x – h = a(y – k)2 • Vertex (h, k) • Focus (h + c, k) • Directrix x = h - c • Axis of Symmetry y = k

  29. Example 3 Find the vertex, focus, directrix , and axis of symmetry of the parabola: y2 – 12x -2y + 25 = 0

  30. Example 4 Find an equation of the parabola that has vertex (4,2) and directrix y = 5

  31. Section 9-4 Ellipses

  32. Ellipse • The set of all points P in the plane such that the sum of the distances from P to two fixed points is a given constant.

  33. Focus (foci) • Each fixed point • Labeled as F1 and F2 • PF1 and PF2 are the focal radii of P

  34. Ellipse- major x-axis

  35. Ellipse- major y-axis

  36. Example 1 • Find the equation of an ellipse having foci (-4, 0) and (4, 0) and sum of focal radii 10. Use the distance formula.

  37. Example 1 - continued • Set up the equation PF1 + PF2 = 10 √(x + 4)2 + y2 + √(x – 4)2 + y2 = 10 • Simplify to get x2 + y2 = 1 25 9

  38. Graphing • The graph has 4 intercepts • (5, 0), (-5, 0), (0, 3) and (0, -3)

  39. Symmetry • The ellipse is symmetric about the x-axis if the denominator of x2 is larger and is symmetric about the y-axis if the denominator of y2 is larger

  40. Center • The midpoint of the line segment joining its foci

  41. General Form x2 + y2 = 1 a2 b2 The center is (0,0) and the foci are (-c, 0) and (c, 0) where b2 = a2 – c2 focal radii = 2a

  42. General Form x2 + y2 = 1 b2 a2 The center is (0,0) and the foci are (0, -c) and (0, c) where b2 = a2 – c2 focal radii = 2a

  43. Finding the Foci • If you have the equation, you can find the foci by solving the equation b2 =a2 – c2

  44. Example 2 Graph the ellipse 4x2 + y2 = 64 and find its foci

  45. Example 3 • Find an equation of an ellipse having x-intercepts √2 and - √2 and y-intercepts 3 and -3.

  46. Example 4 • Find an equation of an ellipse having foci (-3,0) and (3,0) and sum of focal radii equal to 12.

  47. Section 9-5 Hyperbolas

  48. Hyperbola • The set of all points P in the plane such that the difference between the distances from P to two fixed points is a given constant.

  49. Focal (foci) • Each fixed point • Labeled as F1 and F2 • PF1 and PF2 are the focal radii of P

  50. Example 1 • Find the equation of the hyperbola having foci (-5, 0) and (5, 0) and difference of focal radii 6. Use the distance formula.

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