Download presentation

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

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google