1. Chapter 9
Analytic Geometry

2. Distance and Midpoint Formulas
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)

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. The circle with center (h,k) and radius r has the equation
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 x - h = a(y-k)2
Vertex (h,k) symmetry x = h
x - h = a(y-k)2
Vertex (h,k) symmetry y = k

21. (h,k) is the vertex of the parabola
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

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.

51. Example 1 - continued Set up the equation PF1 - PF2 = ± 6
√(x + 5)2 + y2 - √(x – 5)2 + y2 = ± 6
Simplify to get x2 - y2 = 1

52. Graphing The graph has two x-intercepts and no y-intercepts
(3, 0), (-3, 0)

53. Asymptote(s)
Line(s) or curve(s) that approach a given curve arbitrarily, closely
Useful guides in drawing hyperbolas

54. Center
Midpoint of the line segment joining its foci

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

56. Asymptote Equations
y = b/a(x) and
y = - b/a(x)

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

58. Asymptote Equations
y = a/b(x)
and
y = - a/b(x)

59. Example 2
Find the equation of the hyperbola having foci (3, 0) and (-3, 0) and difference of focal radii 4. Use the distance formula.

60. Example 3 Find an equation of the hyperbola with asymptotes
y = 3/4x and y = -3/4x and foci (5,0) and (-5,0)

61. Section 9-6
More on Central Conics

62. Ellipses with Center (h,k)
Horizontal major axis: (x –h)2 + (y-k)2 = 1
a b2
Foci at (h-c,k) and (h + c,k) where c2 = a2 - b2

63. Ellipses with Center (h,k)
Vertical major axis:
(x –h)2 + (y-k)2 = 1
b a2
Foci at (h, k-c) and (h,c +k) where c2 = a2 - b2

64. Hyperbolas with Center (h,k)
Horizontal major axis: (x –h)2 - (y-k)2 = 1
a b2
Foci at (h-c,k) and (h + c,k) where c2 = a2 + b2

65. Hyperbolas with Center (h,k)
Vertical major axis:
(y –k)2 - (x-h)2 = 1
a b2
Foci at (h, k-c) and (h, k+c) where c2 = a2 + b2

66. Example 1
Find an equation of the ellipse having foci (-3,4) and (9, 4) and sum of focal radii 14.

67. Example 2 Find an equation of the hyperbola having foci
(-3,-2) and (-3, 8) and difference of focal radii 8.

68. Example 3 Identify the conic and find its center and foci, graph.
x2 – 4y2 – 2x – 16y – 11 = 0