# Analytic%20Geometry - PowerPoint PPT Presentation

Title:

## Analytic%20Geometry

Description:

### Chapter 9 Analytic Geometry Example 1 Find an equation of the ellipse having foci (-3,4) and (9, 4) and sum of focal radii 14. Example 2 Find an equation of the ... – PowerPoint PPT presentation

Number of Views:136
Avg rating:3.0/5.0
Slides: 69
Provided by: FLS2
Category:
Tags:
Transcript and Presenter's Notes

Title: Analytic%20Geometry

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 v(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)
• (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
• 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
1. a 1/4c where c is the distance between the
vertex and focus
2. Parabola opens upward if agt0, and downward if alt
0

26
y k a(x h)2
1. Vertex (h, k)
2. Focus (h, kc)
3. Directrix y k c
4. Axis of Symmetry x h

27
x - h a(y k)2
1. a 1/4c where c is the distance between the
vertex and focus
2. Parabola opens to the right if agt0, and to the
left if alt 0

28
x h a(y k)2
1. Vertex (h, k)
2. Focus (h c, k)
3. Directrix x h - c
4. 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
• v(x 4)2 y2 v(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

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

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 v2 and - v2 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
• v(x 5)2 y2 - v(x 5)2 y2 6
• Simplify to get x2 - y2 1
• 9 16

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
• a2 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
• b2 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
• a2 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
• a2 b2
• Foci at (h, k-c) and (h, kc) 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