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)
  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

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Section 9-3

• Parabolas

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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

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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

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Equation of a Parabola

• Remember
• y k a(x h)2
• (h,k) is the vertex of the parabola

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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.

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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

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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

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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

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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

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Example 3

• Find the vertex, focus, directrix , and axis of
  symmetry of the parabola
• y2 12x -2y 25 0

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Example 4

• Find an equation of the parabola that has vertex
  (4,2) and directrix y 5

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Section 9-4

• Ellipses

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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.

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Focus (foci)

• Each fixed point
• Labeled as F1 and F2
• PF1 and PF2 are the focal radii of P

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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)

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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

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Center

• The midpoint of the line segment joining its foci

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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 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

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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.

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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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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

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Graphing

• The graph has two x-intercepts and no
  y-intercepts
• (3, 0), (-3, 0)

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Asymptote(s)

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

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Center

• Midpoint of the line segment joining its foci

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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

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Asymptote Equations

• y b/a(x) and
• y - b/a(x)

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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

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Asymptote Equations

• y a/b(x)
• and
• y - a/b(x)

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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.

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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

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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

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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.

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Example 2

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

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Example 3

• Identify the conic and find its center and foci,
  graph.
• x2 4y2 2x 16y 11 0