PreCal Q1 Introduction to Trigonometry and Unit Circle.pptx

Physical Science - Wave Properties of Light.pptx

1. INTRODUCTION TO CONIC SECTIONS PRE-CALCULUS STEM 11

2. LEARNING OBJECTIVES At the end of this lesson, you are expected to: illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases define a circle determine the standard form of equation of a circle

3. TYPES OF CONIC SECTIONS 01

4. CONE A cone is defined as a distinctive three-dimensional geometric figure with a flat and curved surface pointed towards the top .

5. DOUBLE-NAPPED CONE A double-napped cone has two cones connected at the vertex .

6. ACTIVITY: DRAW THAT CURVE

7. CONIC SECTIONS Conic sections are the curves obtained from the intersection between a double-napped cone and a plane .

8. CIRCLE Circles are formed when the intersection of the plane is perpendicular to the axis of revolution. CONIC SECTIONS

9. ELLIPSE Ellipses are formed when the plane intersects one cone at an angle other than 90°. CONIC SECTIONS

10. PARABOLA Parabolas are formed when the plane is parallel to the generating line of one cone. CONIC SECTIONS

11. HYPERBOLA Hyperbolas are formed when the plane is parallel to the axis of revolution or y-axis. CONIC SECTIONS

12. DEGENERATE CONIC SECTIONS Degenerate conic sections are formed when a plane intersects the cone in such a way that it passes through the apex.

13. DEGENERATE CONIC SECTIONS Degenerate conic sections are formed when a plane intersects the cone in such a way that it passes through the apex.

14. INTERACTIVE 3D OF CONIC SECTIONS https://www.intmath.com/plane-analytic-geometry/conic-sections-summary-interactive.php

15. COMMON PARTS OF THE CONIC SECTIONS The conic sections may have looked different; however, they still have common parts. VERTEX CENTER FOCUS DIRECTRIX

16. COMMON PARTS OF THE CONIC SECTIONS VERTEX Vertex is an extreme point on a parabola, hyperbola, and ellipse. Although, ellipse has vertices and co-vertices. with horizontal axis

17. COMMON PARTS OF THE CONIC SECTIONS VERTEX Vertex is an extreme point on a parabola, hyperbola, and ellipse. Although, ellipse has vertices and co-vertices. with vertical axis

18. COMMON PARTS OF THE CONIC SECTIONS FOCUS AND DIRECTRIX The focus and directrix are the point and the line on a conic section that are used to define and construct the curve respectively. The distance of any point on the curve from the focus to the directrix is proportion as shown on the images below by the green lines. In a plane, the circle has no defined directrix. with horizontal axis

19. COMMON PARTS OF THE CONIC SECTIONS FOCUS AND DIRECTRIX The focus and directrix are the point and the line on a conic section that are used to define and construct the curve respectively. The distance of any point on the curve from the focus to the directrix is proportion as shown on the images below by the green lines. In a plane, the circle has no defined directrix. with vertical axis

20. COMMON PARTS OF THE CONIC SECTIONS CENTER For circles , the center is the point equidistant from any point on the surface .

21. Looking at the Cartesian plane, the vertex is at the point (0,0) since it is the extreme point of the parabola. The focus is at (3,0). In order to solve for the directrix , we need to look at the orientation of the parabola. Since the formula has a horizontal axis, the formula for the directrix will be 𝑥 = −𝑐 . Thus, the equation of the directrix is 𝑥 = −3.

22. LET’S TRY Given the curve on the Cartesian plane below, identify the focus, vertex, and directrix.

23. Looking at the graph, we can see that the foci are at (−2 − 3) and (6, −3). Note that the center is the midpoint of the foci. Thus, we have the following solution:

24. LET’S TRY Identify the coordinates of the foci and the center of the graph below.

25. Step 1: Plot the vertex and the focus, and identify the curve. Looking at the Gateway Arch, it looks like a parabola. Step 2: Solve for the directrix. Since the focus is at (0, −3) , the directrix is 𝑦 = 3

26. ASSESSMENT

27. A. Identify the conic section or the part that is being described. 1. These are the conic sections that are formed when the plane intersects the double-napped cone in such a way that it passes through the apex. 2. This conic section is formed when the plane is parallel to the axis of revolution. 3. It is the midpoint of the two foci for ellipse and hyperbola. 4. It refers to the extreme point of a parabola. 5. These are the curves that are obtained between the intersection of a double-napped cone and a plane. ASSESSMENT

28. B.Using the image below, complete the table and solve for the directrix given the vertices and foci.

29. Assignment: Challenge Yourself Answer the following questions: 1. You saw Albert playing with a double-napped cone and a paper. He put the paper on top of one cone and said that he was able to form a conic section. Do you agree with him? Explain your answer. 2. A glass was placed on the table. If you hold a flashlight as shown below, what kind of curve will be formed by its shadow? 3. If you shift a parabola with vertex at the origin, two units to the right, what will be the new vertex?

30. CIRCLES 02

31. CIRCLE A circle is formed when a plane perpendicular to the axis intersects a double-napped cone. RECALL

32. ARCHITECTURE Circular shapes are mostly used as symbolic designs in architecture around the world. Moreover, the use of circles is more efficient when it comes to savings in surface area . It also has better behavior regarding winds and solar radiation. APPLICATIONS OF CIRCLES

33. TRANSPORTATION Wheels made it easier for people to travel great distances . This invention has been one of the most useful and essential of all times. Also, when determining distances, GPS heavily depends on circles . Using circle theories, it calculates distances between satellites and points. APPLICATIONS OF CIRCLES

34. PHOTOGRAPHY Adjusting camera lenses is done by moving the lenses in a screw-like manner . This is the reason why camera lenses are circular in shape . It is easier for the photographers and videographers to focus and adjust the zoom lenses or the focal lengths of their cameras. APPLICATIONS OF CIRCLES

35. The set of points in a plane that are all equidistant from a given point , called the center, forms a circle. Any segment with endpoints at the center and a point on the circle is a radius of the circle.

36. Suppose that (𝑥,𝑦) are the coordinates of a point on the circle. Moreover, the center of the circle with radius 𝑟 is at (ℎ,𝑘). It follows that the value of 𝑟 is equal to the distance between (𝑥,𝑦) and (ℎ,𝑘). EQUATION OF A CIRCLE IN STANDARD FORM

37. How do we get the radius? EQUATION OF A CIRCLE IN STANDARD FORM

38. Suppose that (𝑥,𝑦) are the coordinates of a point on the circle. Moreover, the center of the circle with radius 𝑟 is at (ℎ,𝑘). It follows that the value of 𝑟 is equal to the distance between (𝑥,𝑦) and (ℎ,𝑘). This may be calculated using the distance formula . EQUATION OF A CIRCLE IN STANDARD FORM

39. EQUATION OF A CIRCLE IN STANDARD FORM DISTANCE FORMULA

40. EQUATION OF A CIRCLE IN STANDARD FORM

41. EQUATION OF A CIRCLE IN STANDARD FORM This equation is known as the standard form of equation of a circle with center at (ℎ,𝑘) and radius 𝑟. This is also known as the center-radius form .

42. EQUATION OF A CIRCLE IN STANDARD FORM If a circle has the center at the origin (0,0) and has radius 𝑟: This is the standard form of equation of a circle with center at the origin .

43. EXAMPLE 1 Find the equation of the circle with center at the origin and has a radius of 10 units.

44. EXAMPLE 1 Find the equation of the circle with center at the origin and has a radius of 10 units.

45. EXAMPLE 1 Find the equation of the circle with center at the origin and has a radius of 10 units.

46. EXAMPLE 1 Find the equation of the circle with center at the origin and has a radius of 10 units.

47. LET’S TRY Find the equation of the circle with center at the origin and has a radius of 12 units.

48. EXAMPLE 2 Find the equation of the circle with center at (−3,−1) and has a radius of √6 units.

49. EXAMPLE 2 Find the equation of the circle with center at (−3,−1) and has a radius of √6 units.

50. EXAMPLE 2 Find the equation of the circle with center at (−3,−1) and has a radius of √6 units.

51. EXAMPLE 2 Find the equation of the circle with center at (−3,−1) and has a radius of √6 units.

52. LET’S TRY Find the equation of the circle with center at (0,−5) and has a radius of √10 units.

53. We have learned in the first two examples how to determine the equation of a circle given its center and radius. What if we reverse the process?

54. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

55. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

56. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

57. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

58. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

59. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

60. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

61. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

62. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

63. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

64. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

65. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

66. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

67. EXAMPLE 3 Find the center and the radius of the circle whose equation is (𝑥−10) 2 +(𝑦+8) 2 =49.

68. LET’S TRY Find the center and radius of the circle whose equation is (𝑥+1) 2 +(𝑦−5) 2 = 20.

69. EQUATION OF A CIRCLE IN GENERAL FORM where the numerical coefficients are real numbers and 𝐴=𝐵. Moreover, both 𝐴 and 𝐵 cannot be zero at the same time.

70. EXAMPLE 4 Identify the center and the radius of the circle defined by the equation 𝑥 2 +𝑦 2 +4𝑥−6𝑦+3=0.

71. EXAMPLE 4 Identify the center and the radius of the circle defined by the equation 𝑥 2 +𝑦 2 +4𝑥−6𝑦+3=0.

72. EXAMPLE 4 Identify the center and the radius of the circle defined by the equation 𝑥 2 +𝑦 2 +4𝑥−6𝑦+3=0.

73. EXAMPLE 4 Identify the center and the radius of the circle defined by the equation 𝑥 2 +𝑦 2 +4𝑥−6𝑦+3=0.

74. EXAMPLE 4 Identify the center and the radius of the circle defined by the equation 𝑥 2 +𝑦 2 +4𝑥−6𝑦+3=0. The center is at (−𝟐,𝟑) and the radius is √𝟏𝟎.

75. LET’S TRY Identify the center and the radius of the circle defined by the equation 𝑥 2 +𝑦 2 +2𝑥+2𝑦−7=0.

76. ALTERNATIVE WAY Note: These formulas work when the value of 𝐴 and 𝐵 are both 1.

77. EXAMPLE 5 Find the general form of equation of the circle illustrated below.

78. EXAMPLE 5

79. EXAMPLE 5

80. EXAMPLE 5

81. EXAMPLE 5 Therefore, the general form of the equation of the circle is 𝒙 𝟐 +𝒚 𝟐 −𝟖𝒙−𝟒𝒚+𝟏𝟏=𝟎 .

82. LET’S TRY Find the general form of equation of the circle illustrated.

83. EXAMPLE 6 Rowell’s house has a portable Wi-Fi router that can reach a field of about 50 feet from its location. Suppose their neighborhood represents the Cartesian plane, his location is in the origin, and his house is situated 30 feet north and 10 feet east from where he is. a. Find the equation of the circle in general form which describes the boundary of the Wi-Fi signal. b. Determine whether he can still connect to their Wi-Fi at home.

84. EXAMPLE 6

85. EXAMPLE 6

86. EXAMPLE 6

87. LET’S TRY A cellular network company uses towers to transmit communication information. A tower located at (−1,−4) of the company grid can transmit signals up to a 7-kilometer radius. Find the general form of equation of the boundary this tower can transmit signals to.

88. ASSESSMENT

89. C. Analyze and solve the problem below. The Pampanga Eye currently holds the title for the tallest Ferris wheel in the Philippines. It is situated in Sky Ranch Pampanga, a theme park in San Fernando City. The Ferris wheel is 50 meters in diameter and has a height of 65 meters. Find an equation for the wheel, assuming that its center lies on the 𝑦-axis and that the ground is the 𝑥-axis. ASSESSMENT

90. FAST WAY TO GRAPH CONIC SECTIONS USING SOFTWARE TOOLS https://www.symbolab.com/solver/conic-sections-calculator

91. EXAMPLE Graph the equation 𝑥 2 +𝑦 2 =36.

92. Graph the equation 𝑥 2 +𝑦 2 =36.

93. Graph the equation 𝑥 2 +𝑦 2 =36.

94. Graph the equation 𝑥 2 +𝑦 2 =36.

95. EXAMPLE Graph the equation (𝑥−2) 2 +(𝑦−3) 2 =16.

96. Graph the equation (𝑥−2) 2 +(𝑦−3) 2 =16.

97. Graph the equation (𝑥−2) 2 +(𝑦−3) 2 =16.

98. Graph the equation (𝑥−2) 2 +(𝑦−3) 2 =16.

99. Graph the equation (𝑥−2) 2 +(𝑦−3) 2 =16.

100. EXAMPLE Graph the circle with center at (−3,−1) and tangent to the 𝑦-axis.

101. DID YOU REMEMBER?

102. Graph the circle with center at (−3,−1) and tangent to the 𝑦-axis.

103. Graph the circle with center at (−3,−1) and tangent to the 𝑦-axis.

104. Graph the circle with center at (−3,−1) and tangent to the 𝑦-axis.

105. PERFORMANCE TASK

106. A. Graph the circle with the given center and radius. 1. (1,3); 𝑟=2 2. (−2,2); 𝑟=3 B. Graph the circle with the given equation in standard form. 3. (𝑥−2) 2 +(𝑦+3) 2 =36 C. Graph the circle with the given equation in general form. 4. 𝑥 2 +𝑦 2 +4𝑥−6𝑦+12=0 D. Sketch the graph and find the equation in standard form of the circle being described in each item. 5 . a circle with center at (3,4) and tangent to the 𝑥-axis PERFORMANCE TASK 2