Chapter 2: Circle. Mr. Migo M. Mendoza. SSMth1: Precalculus. Science and Technology, Engineering and Mathematics (STEM)

Transcription

1 Chapter 2: Circle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

2 Chapter 2: Circles Lecture 2: Introduction to Circles Lecture 3: Form of a Circle Lecture 4: General Form and Standard Form of a Circle Lecture 5: Circles Determined by Different Conditions Lecture 6: Tangent to a Circle

3 Nice to Know: TED Ed Video: Why are Manholes Cover Round? by Marc Chamberland

4 From TED Ed Video: Keep your eyes open and you just might come across a rule of triangle manhole.

5 Nice to Know: Why is πr 2 the area of a circle?

6 Lecture 2: Introduction to Circle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

7 What is a Circle?

8 Definition of Circle A circle is a set of all points ( x, y) in a plane whose distances from a fixed point is a constant.

9 The Center and the Radius The fixed point is called the CENTER, and the distance from the center to any point of the circle is referred to as the RADIUS.

10 The Derivation of Equation of a Circle

11 Something to think about If the radius is the distance from a fixed point to any point on the circle, what formula we have learned from junior high school can we use to derive the standard equation of a circle?

12 The Distance Formula: d 2 2 x x y y

13 Form of Circle # 1: The Standard Form of an Equation of a Circle with Radius r and Center at (h, k): x h2 y k2 r 2

14 Something to think about What will happen to the standard equation of a circle with radius r if the center is at the origin?

15 Form of Circle # 2: The Standard Form of an Equation of a Circle with Radius r and Center at the origin (0, 0): x 2 y 2 r 2

16 Something to think about What will happen to the standard equation of a circle with the center at the origin if the radius is 1?

17 Form of Circle # 3: The Standard Form of an Equation of a Circle with Radius 1 and Center at the origin (0, 0): x 2 y 2 1

18 Did you know? In addition, the standard form of an equation of the circle with radius 1 and center at the origin is called UNIT CIRCLE and has the equation: x 2 y 2 1

19 Form of Circle #4: The General Form of the Equation of a Circle is: x 2 y 2 Dx Ey F 0

20 Lecture 3: Forms of a Circle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

21 Learning Expectation: This lecture will discuss how to set up the graph of a circle and determine its radius.

22 Example 5: Graph the circle with standard form of: x 2 y 2 16

23 Something to think about What do you know about radii of the same circle?

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25 Example 6: Determine the general equation of the circle whose center is (3, -1) and whose graph contains the point (7, -1). Also, sketch the graph.

26 Did you know? We can convert standard equation of a circle to its general form by expanding the binomials using the FOIL method.

27 Final Answer: Therefore, the general equation of the circle whose center is (3, -1) and whose graph contains the point (7, -1) is: x 2 y 2 6x 2 y 6 0

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29 Example 7: Find the general equation of the circle whose center is (2, 6) and whose radius is 3. Also, please graph the circle.

30 Final Answer: Thus, the general equation of the circle whose center is (2, 6) and whose radius is 3: x 2 y 2 4x 12 y 31 0

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32 Performance Task 2: Please download, print and answer the Let s Practice 2. Kindly work independently.

33 Lecture 4: Converting General Form to Standard Form of a Circle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

34 A Short Recap: What is the standard form of the equation of the circle where the center is C (h, k) and the radius is r?

35 A Short Recap: What is the general form of the equation of a circle?

36 Something to think about What benefit can we get from using standard equation instead of general equation of a circle?

37 Did you know? The standard form of the circle is more convenient in the sense that we can easily identify the center and the radius of a circle.

38 Example 8: Change the general equation of a circle, x 2 y 2 8x 6 y 0 to standard form and determine the center and the radius. Also, please sketch the graph.

39 Did you know? To convert general equation of a circle to its standard equation, use the completing the square method.

40 Step 1: Completing the Square Group the equation according to its variables.

41 Step 2: Completing the Square Ensure that the coefficients of x 2 and y 2 are both 1.

42 Step 3: Completing the Square Take the half of the coefficient of x and y, squared it, and add it to both sides of the equation.

43 Step 4: Completing the Square Factor by perfect square trinomial.

44 Step 5: Completing the Square Simplify the value of the radius (r 2 ).

45 Final Answer: Thus, the standard form of the equation of the circle is 2 2 x 4 y 3 25, while the center is at (-4, 3) and r = 5.

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47 Example 9: Express the given general equation below to its standard form: x 2 y 2 4x 8y 20 0

48 Something to think about Is there an easiest way to convert the general form of a circle to its standard form?

49 Classroom Task 3: By completing the square, derive the equations which we can use to easily convert a circle in general form to its standard form.

50 Thus, center C (h, k) and radius r is equivalent to: The center C (h, k) and radius r can be obtained using the following formula: 2 D h 2 E k F E D r F E D r

51 Something to think about The center C (h, k) is (2, 4). Also, note that the right side of the equation is zero. So, what can you conclude about this circle?

52 Point Circle or Degenerate Circle Thus, point (2, 4) is the only point on a plane that satisfies the equation and the radius is zero. This type of equation is referred as POINT CIRCLE or a DEGENERATE CIRCLE.

53 Example 10: Express the general form of a circle below to its standard form: x 2 y 2 6x 10 y 40 0

54 Something to think about Revisit our previous examples, what have you observed on the value of the radius r 2? What can you conclude?

55 Take Note: The value of r 2 in the standard equation of a circle is always positive. If r 2 is negative, then the solution does not exist.

56 Take Note: Note that the right side of the equation is negative. This implies that there is no point in the plane that satisfies the equation 2 2 x y 6x Therefore, the circle DOES NOT EXIST. y 0

57 Something to think about Why do you think the value of radius r 2 will never ever be negative?

58 To sum it up What conclusions can we make with respect to the radius of circle?

59 Conclusion Number 1: Whenever the radius of a circle is a positive value, the circle exists.

60 Conclusion Number 2: Whenever the radius of a circle is exactly equal to zero, the circle is a point or degenerate circle.

61 Conclusion Number 3: Whenever the radius of a circle is a negative value, the circle does not exist.

62 Performance Task 3: Please download, print and answer the Let s Practice 3. Kindly work independently.

63 Lecture 5: Circles Determined by Different Conditions SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

64 What should you expect? This section illustrates how to establish the equation of a circle given different conditions. Three of the possible cases are presented in the succeeding examples.

65 Example 11: Determine the general equation of the circle which passes through the points P 1 (-1, 2), P 2 (0, 5), and P 3 (2, 1). Also, please sketch the graph.

66 Something to think about Where can we find the given three points on the circle?

67 Method Number 1: Method Number 1: Determining the GENERAL EQUATION given three points on the circumference of the circle.

68 Step 1: Since we already have values for (x, y), out of it we will construct three (3) equations which are of the form: x 2 y 2 Dx Ey F 0

69 Something to think about What have you observed on our three equations? In order for us to find the values of D, E, and F, what method we have learned from junior high school can we use?

70 Step 2: Through the use of solving systems of linear equation (elimination method), we will combine the first and second equation so we can eliminate variable(s).

71 Step 3: Through the use of solving systems of linear equation (elimination method), we will combine the first and third equation so we can eliminate variable(s).

72 Step 4: Combine the fourth and the fifth equations and solve for the value of E.

73 Step 5: Substitute E = -6 to the fourth equation to obtain the value of D.

74 Step 6: To obtain the value of F, we substitute also E = -6.

75 Step 7: Lastly, we substitute D = -2, E = -6, and F = 5 to the general equation of a circle.

76 Method Number 2: Method Number 2: Determining the STANDARD EQUATION given three points on the circumference of the circle.

77 Final Answer: Thus, the general equation of the circle is x 2 y 2 2x 6 y 5 0 which contains the points P 1 (-1, 2), P 2 (0, 5), and P 3 (2, 1).

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79 Example 12: Determine the equation of the circle passing through P 1 (4, 0), and P 2 (3, 5), with a line 3x 2y 7 0 passing through the center. Also, please sketch the graph.

80 Take Note: If 3x 2y 7 0 is a line, by definition 3x 2y 7 0 is a collection of infinitely many points. Moreover, if 3x 2y 7 0 passes through the center, and remember by definition, the center is a fixed point, therefore the center of the circle is one of the infinitely many points of 3x 2y 7 0.

81 Also If 3x 2y 7 0 contains the fixed point C (h, k), the coordinates of the center are solutions to the line 3x 2y 7 0. Since its solutions are denoted by x and y, we will let C (h, k) be C (x, y) as the center of the circle.

82 Final Answer: Therefore, the equation of the circle passing through P 1 (4, 0), and P 2 (3, 5), with a line 3x 2y 7 0 through the center is: passing x 2 y 2 2x 4 y 8 0

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84 Example 13: Find the equation of the circle that circumscribe the triangle determined by the lines x = 0, y = 0 and 3x 4y Also, please sketch the graph.

85 Something to think about When can we say a triangle is circumscribed by a circle?

86 Tell Me: Which is Which? From the two figures, which is circumscribed circle and inscribed circle?

87 Circumscribed Circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called circumcenter and its radius is called the circumradius.

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89 Performance Task 4: Please download, print and answer the Let s Practice 4. Kindly work independently.

90 Lecture 6: Tangent to a Circle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

91 Did you know? There are three (3) saddest love stories in Mathematics

92 The Painful Asymptote There are people who may get closer and closer to one another, but will never be together.

93 The Painful Parallel You may encounter potential people, bump onto them, see them from afar, but will never actually get to know and meet them; even in the longest time.

94 The Painful Tangent Some people are only meant to meet one another at one point in their lives, but are forever parted.

95 Something to think about What are the three (3) conditions that guarantee a line is tangent to a circle?

96 Definition: Tangent to a Circle A line in the plane of the circle that intersects the circle at exactly one point is called tangent line. The point of intersection is called the point of tangency.

97 The Tangent-Line Theorem:

98 The Tangent-Line Theorem If a line is tangent to a circle, then it is perpendicular to the radius at its outer endpoint.

99 The Tangent-Line Theorem:

100 Something to think about If tangent is a line, what kind of function is a tangent line? Also, what s its general equation?

101 Did you know? A tangent is associated to a graph of a line. Thus, a tangent is a linear function which has a general equation of: Ax By C 0

102 Did you know? There are four main types of problems concerning tangents to circles.

103 The Four Main Types 1. Tangent at a Given Point 2. Tangent in Prescribed Direction 3. Inscribed Circle in a Triangle 4. Tangents from a Point Outside the Circle

104 Tangent to a Circle: Tangent at a Given Point SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

105 Example 14: Given the equation of the circle x 2 y 2 8x 14 y Prove that x 2y 0 is a tangent to the circle. Sketch the graph.

106 Something to think about How can we prove that x 2y 0 is a tangent to the 2 2 circle x y 8x 14 y 45 0?

107 Analysis: If x 2y 0 is a tangent line to the circle 2 2 x y 8x 14 y 45 0, it touched the at exactly one point, and that point is what we call the point of tangency. Moreover, since x 2y 0 is a tangent line, it is a collection of infinitely many points.

108 Analysis: x If is tangent to the circle the point of tangency PT (x, y) is one of the infinitely many points of. Also, the point of tangency is on the circumference of the circle 2 y 8x 14 y Hence, it is one of the infinitely many points in the circumference of the 2 2 circle. 2 x 2y 0 x 2y 0 x y 8x 14 y 45 0

109 Analysis: x 2y 0 Furthermore, if the tangent line and 2 2 the circumference of the circle x y 8x 14 y both contain the point of tangency PT (x, y). Therefore, the PT (x, y) is the solution to both the x 2y 0 tangent line and the 2 2 circle x y 8x 14 y 45 0.

110 Proving a Line is Tangent to a Circle We need to show that x 2y 0 touches the circle with equation. 2 2 x y 8x 14 y 45 0 in a single point. This single point is a common point of the tangent and the circle. Thus, it is the solution to the equation of the circle and the tangent line.

111 Take Note: If a line touches the circle in a single point, then it s a tangent.

112 Take Note: If a line touches the circle in two points, then it s a secant.

113 Take Note: If a line does not touch the circle there is no solution.

114 Final Answer: Thus, the point of tangency is at (6, 3). Since there is only one solution, this shows that the line x 2y 0 just touches the circle in one place and therefore it is a tangent.

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116 Example 15: Find the equation of the tangent line to the circle x 2 y 2 6x 10 y 17 0 at the point (-2, 1). Sketch the graph.

117 Five Forms of Linear Equation: The Slope-Intercept Form y mx b

118 Five Forms of Linear Equation: The Point-Slope Form ( y y ) m( x x 1 1 )

119 Five Forms of Linear Equation: The Two-Point Form y y ( y y ) 2 1 ( x x ) 1 1 x x 2 1

120 Five Forms of Linear Equation: The Intercept Form x a b y 0

121 Five Forms of Linear Equation: The Normal Form x cos y sin p 0

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123 The Tangent-Line Theorem If a line is tangent to a circle, then it is perpendicular to the radius at its outer endpoint.

124 The Perpendicular Slope Theorem: If two lines are perpendicular, having respective slopes, m 1 and m 2, the slope of the line is the negative reciprocal of the slope of the other line: m 2 1." m 1

125 Final Answer: Therefore, the equation of the tangent line is: x 4y 0.

126 Example 16: Find the equation of the circle with center (4, 0) and touching the line 2x y Sketch the graph.

127 Analysis: Observe that the only given we have is the center C (h, k) = C (4, 0) and an equation of a line 2x y If we are to find the equation of the circle we should have the value of radius. However, we all know that we can find the radius by knowing how far the center C (h, k) = C (4, 0) to the tangent line 2x y

128 Something to think about If d ( x y x1 ) ( y2 1) is the formula for finding the distance given TWO POINTS, then what is the formula for finding the distance given a POINT and a LINE?

129 Formula for Finding the Distance Given a Point and a Line: Formula for finding the distance given a point and a line: Ax By C d A 2 B 2

130 Final Answer: Therefore, the equation of the circle with center (4, 0) and touching the line 2x y 18 0 is: x 2 y 2 8x 4 0.

131

132 Tangent to a Circle: Tangent in Prescribed Direction SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

133 What should you Expect? This section will illustrates how to determine the tangents parallel to the circle given an equation of the line which is parallel or perpendicular to the tangent line.

134 Example 17: Determine the lines tangent to the circle x 2 y 2 6x 4y 12 0 and parallel to the line 3x 4y Sketch the graph.

135 Take Note: To determine the equations of the tangent lines which are also parallel to 3x 4y 30 0, we have to use the formula for finding the radius given a point and a line which is Ax By C r. Please take 2 2 A B note that the value of the raduis is ±5 units. Since we are looking for two equations of tangent lines which are also parallel to 3x 4y 30 0, for special case like this we will use the negative radius.

136 Final Answer: Thus, the equations are 3x 4y 26 0 and 3x 4y 24 0.

137

138 Example 18: Given a line circle x 2 y 2 8x x 2y 4 0, and find the equations of the tangents to the circle which are perpendicular to the line. Also, please sketch the graph. 4 0

139 Take Note: We need to find two equations of tangent lines to the 2 2 circle x y 8x 4 0 which are also perpendicular to x 2y 4 0. Take note that using the Perpendicular Slope Theorem which states that, If two lines are perpendicular, having respective slopes, m 1 and m 2, the slope of the line is the negative reciprocal of the slope of the other line, we can actually find the slopes of L 2 denoted by m 2 and L 3 x 2y 4 0 denoted by m 3. If the slope of m 1 = -1/2, hence m 2 = m 3 = 2. is

140 Take Note: As you can observed, we are to find two equations of 2 2 tangent lines to circle x y 8x 4 0 which are also perpendicular to x 2y 4 0 using the Slope-Intercept Form y mx b since we have m 2 = m 3 = 2. Thus, we have to find the y-intercepts b 1 for L 2 and b 2 L 3. To find this, we have to use the formula for finding the distance from a point to the line using the formula: r y mx b mx 1 m b 2 1 y 1

141 Final Answer: Substituting the value of m (slope) and b (y-intercept) in y mx b, the equation of the tangents are 2x y 2 0. and 2x y 18 0.

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143 Tangent to a Circle: Inscribed Circle in a Triangle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

144 Inscribed Circle It is the largest possible circle that can be drawn inside the triangle in which each of the triangles' sides is a tangent to the circle. We also refer to this as INCENTER OF A TRIANGLE.

145 Incenter It is the point at which the angle bisectors of a triangle intersect and it is the center of the circle that can be inscribed in a triangle.

146 Example 19: A triangle has its sides having equation equal to 2x y 16 0 and x 2y 9 Find the equation of a circle inscribed in a triangle. Also, please sketch the graph 2x y 0, 0.

147 Final Answer: Therefore, the equation of the circle tangent to the three given lines is: 5x 2 5y 2 40 x 10 y 36 0.

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149 Tangent to a Circle: Tangents from a Point Outside the Circle SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

150 What should you Expect? This section illustrates how to determine the tangents that contains a point outside the circle.

151 Example 20: Find the equations of the tangent line to the circle x 2 y 2 4x 6y 4 0 from the point (1, 7). Sketch the graph.

152 Final Answer: Therefore, the tangent lines are 7x 24y 161 and x

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154 Performance Task 5: Please download, print and answer the Let s Practice 5. Kindly work independently.