2. Chapter 6
Analytic Geometry

3. 6.1
Parabolas

4. Conic Sections
Parabolas, circles, ellipses, and hyperbolas form a group of curves known as conic sections.

5. Parabola with Horizontal Axis
The parabola with vertex (h, k) and the horizontal line y = k as axis has an equation of the form x  h = a (y  k)2.
The parabola opens to the right if a > 0 and to the left if a < 0.

6. Example: Graph x + 2 = (y  3)2
Vertex = (2, 3)
Opens right a = 1 > 0
Domain [2, )
Range (,) 5 2 1 4 1 3 2 y x

7. Example Graph x = 2y2 + 4y + 6 Vertex = (4, 1) y = 1 Domain [4, )
Range (, )

8. Parabola
A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

9. Parabola with Vertical Axis and Vertex (0, 0)

10. Parabola with Horizontal Axis and Vertex (0, 0)

11. Example: Find the focus, directrix, vertex, and axis of each parabola
Example: Find the focus, directrix, vertex, and axis of each parabola. Then graph the parabola.
a) x2 = 12y
Vertex (0, 0)
Focus (0, 3)
Directrix y = 3
F(0, 3)
y = 3
V(0, 0)

12. Find the focus, directrix, vertex, and axis of each parabola
Find the focus, directrix, vertex, and axis of each parabola. Then graph the parabola.
b) Graph y2 = 12x
Vertex (0, 0)
Focus (3, 0)
Directrix x = 3
F(3, 0)
x = 3

13. Writing Equations of Parabolas
Write an equation for a parabola with focus (5, 0) and vertex at the origin.
Since the focus and vertex are both on the x-axis, the parabola is horizontal. It opens to the right since p is positive. The equation will have the form y2 = 4py.
y2 = 4(5)y or y2 = 20y

14. Equation Forms for Translated Parabolas
A parabola with vertex (h, k) has an equation of the form
where the focus is distance |p| from the vertex.

15. Example
Write an equation for the parabola with vertex (1, 4) and focus (1, 4), and graph it. Give the domain and range.
Solution: The focus is to the left of the vertex, the axis is horizontal and the parabola opens to the left.
The distance between the vertex and focus is (1  (1)) = 2, so p = 2 (since it opens left).

16. Example continued
Equation:
The domain is (, 1]
Range is (, ).

17. 6.2
Ellipses

18. Ellipse
An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is constant. Each fixed point is called a focus (plural, foci) of the ellipse.
Two axis of symmetry, the major axis (the longer one) and the minor axis (the shorter one).
The foci are always located on the major axis.
The endpoints of the major axis are the vertices of the ellipse.

19. Standard Forms of Equations for Ellipses

20. Standard Forms of Equations for Ellipses

21. Example
For the ellipse find the vertices and the foci. Then draw the graph.
a = 7 and b = 6. The major axis is vertical.
Vertices: (0, 7) and (0, 7)

22. Example continued The foci are (0, 3.61) and (0, 3.61).
The x-intercepts are (6, 0) and (6, 0)
We plot the vertices and the x-intercepts and connect the four points with a smooth curve.

23. Example
Write the equation of the ellipse having center at the origin, foci at (0, 4) and (0, 4), and major axis of length 12 units.
Solution: The major axis has length 12 units, 2a = 12 and a = 6.
a = 6, c = 4

24. Example continued The equation of the ellipse is:
The domain of the relation is
The range of the relation is [6, 6]

25. Example Graph Solution: Square both sides to get
the equation of an ellipse with x-intercepts 6 and y-intercepts 3.

26. Ellipse Centered at (h, k)
An ellipse centered at (h, k) with horizontal major axis of length 2a has equation
There is a similar result for ellipses having a vertical major axis.

27. Example Graph Center: (2, 1) a = 5, b = 4
Vertices through (2, 1) up 5 units and down 5 units (2, 6) and (2, 4)
Two other points on the ellipse are 4 units to the right and left of the center. (6, 1) and (2, 1)
Domain: [6, 2]
Range: [4, 6]

28. Conic
A conic is the set of all points P(x, y) in a plane such that the ratio of the distance from P to a fixed point and the distance from P to a fixed line is constant.
The constant ratio is called the eccentricity of the conic.
For the ellipse, eccentricity is a measure of its “roundness.”
The ratio is defined by

29. Example Find the eccentricity of the ellipse. Divide by 50 to obtain
a2 = 10, with
Find c,
Eccentricity

30. Example
The orbit of the planet Mars is an ellipse with the sun at one focus The eccentricity of the ellipse is .0935, and the closest distance that Mars comes to the sun is million mi.
(Source: World Almanac and Book of Facts.)
Find the maximum distance of Mars from the sun.

31. Example continued
Mars is closest to the sun when Mars is at the right endpoint of the major axis and farthest from the sun when Mars is at the left endpoint.
The smallest distance is a  c, and the greatest distance is a + c.
a  c = 128.5, c = a  Using

32. Example continued Then c = 141.8  128.5 = 13.3
and a + c = = 155.1
The maximum distance of Mars from the sun is about million mi.

33. 6.3
Hyperbolas

34. Hyperbola
A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points is constant. The two fixed points are called the foci of the hyperbola.

35. Example
Graph
Sketch the asymptotes, and find the coordinates of the vertices and foci. Give the domain and range.
a = 4 and b = 5
Asymptotes:
If x = 4, then y = 5.
If x = 4, then y = 5.

36. Example continued
Foci:
Domain:
[, 4]  [4, )
Range:
(, )

37. Standard Forms of Equations for Hyperbolas

38. Standard Forms of Equations for Hyperbolas

39. Example
Graph 16x2  25y2 = 400
The hyperbola is centered at the origin, has foci on the x-axis, and has vertices (5, 0) and (5, 0).
Asymptotes:

40. Example Graph Center = (1, 3) a = 2 b = 3 Vertices: (1, 1) and (1, 5)
Asymptotes:

41. Eccentricity Narrow hyperbolas have e near 1
Wide hyperbolas have large e.

42. Example
Find the eccentricity of the hyperbola
a2 = 16; thus a = 4,

43. Summary of Eccentricity
and e > 1
Hyperbola
and 0 < e < 1
Ellipse
e = 0
Circle
e = 1
Parabola
Eccentricity
Conic

44. 6.4
Summary of Conic Sections

45. Characteristics
The special characteristics of the equation of each conic section is summarized in the table.
x2  y2 = 1
AC < 0
Hyperbola
A  C, AC > 0
Ellipse
x2 + y2 = 16
A = C  0
Circle
x2 = y + 4
(y  2)2 = (x + 3)
Either A = 0 or C = 0 but not both
Parabola
Example
Characteristic
Conic Section

46. Summary of Conic Sections--Parabola
Equation
y  k = a(x  h)2
Description
Open up if a > 0
Down if a < 0
Vertex is (h, k)
Identification
x2 term
y is not squared

47. Summary of Conic Sections--Parabola
Equation
x  h = a(y  k)2
Description
Open right if a > 0
Open left if a < 0
Vertex is (h, k)
Identification
y2 term
x is not squared

48. Summary of Conic Sections--Circle
Equation
(x  h)2 + (y  k)2 = r2
Description
Center is (h, k)
Radius is r
Identification
x2 and y2 terms have the same positive coefficient.

49. Summary of Conic Sections--Ellipse
Equation
Description
x-intercepts are a and a
y-intercepts are b and b
Identification
x2 and y2 terms have different positive coefficients

50. Summary of Conic Sections--Ellipse
Equation
Description
x-intercepts are b and b
y-intercepts are a and a
Identification
x2 and y2 terms have different positive coefficients

51. Summary of Conic Sections--Hyperbola
Equation
Description
x-intercepts are a and a
Asymptotes are found from (a, b), (a, b), (a, b), and (a, b).
Identification
x2 term has a positive coefficient.
y2 term has a negative coefficient.

52. Summary of Conic Sections--Hyperbola
Equation
Description
y-intercepts are a and a
Asymptotes are found from (b, a), (b, a), (b, a), and (b, a).
Identification
y2 term has a positive coefficient.
x2 term has a negative coefficient.

53. Examples
Determine the type of conic section represented by each equation, and graph it.
a) x2 = y2
b) x2  2x + y2 + 6y = 6
c) 16x2  96x + 9y2 + 18y = 9
d) x2  4x + 2y + 2 = 0

54. x2 = y2
Solution:
The equation represents a hyperbola with center at the origin.
Asymptotes:
x-intercepts = 6

55. x2  2x + y2 + 6y = 6
Complete the square.

56. x2  2x + y2 + 6y = 6 continued
Equation of a circle with center (1, 3) and radius 2. If we had obtained a negative number on the right side, the equation would have no solution.

57. 16x2  96x + 9y2 + 18y = 9
The equation could represent an ellipse but not a hyperbola since both x2 and y2 terms are positive.
Complete the square.

58. 16x2  96x + 9y2 + 18y = 9 continued
The equation represents an ellipse centered at
(3, 1).

59. x2  4x + 2y + 2 = 0
Since only one variable is squared the equation represents a parabola. Get the y term alone.

60. x2  4x + 2y + 2 = 0 continued
The parabola has vertex (2, 1) and opens down.

61. 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
[distance of P from F] = e • [distance of P from L]
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.