Precal Lesson 2 Circles lesson in mathematics
- 1. Precalculus Science, Technology, Engineering, and Mathematics Lesson 1.2 Definition and Equation of a Circle
- 2. 2 Have you ridden a Ferris wheel before? One distinguishable fact about this ride is that it is circular in shape and the points along the outer rim of the wheel have equal distances from the center.
- 3. Learning Competencies At the end of the lesson, you should be able to do the following: 3 ● Define a circle (STEM-PC11AG-1a-2). ● Determine the standard form of equation of a circle (STEM-PC11AG-1a-3).
- 4. Learning Objectives At the end of the lesson, you should be able to do the following: 4 ● Define a circle. ● Determine the equation of a circle given its center and radius and vice versa. ● Convert the general equation of a circle into its standard form and vice versa. ● Solve situational problems involving circle.
- 5. 5 When can we say that a figure is a circle?
- 6. 6 Recall that a circle is formed when a plane perpendicular to the axis intersects a double-napped cone. Circle
- 7. 7 The set of points in a plane, which are all equidistant from a given point, called the center, forms a circle. Circle center
- 8. 8 Any segment with endpoints at the center and a point on the circle is a radius of the circle. Circle radius 𝑨 𝑪
- 9. 9 Like all the other graphs in the Cartesian plane, a circle may be represented by an equation. Circle
- 10. 10 How do you represent the equation of a circle?
- 11. 11 Any segment with endpoints at the center and a point on the circle is a radius () of the circle. Equation of a Circle in Standard Form
- 12. 12 Given the coordinates of a point on the circle as and the center of the circle at may be calculated using the distance formula. Equation of a Circle in Standard Form
- 13. 13 Squaring both sides of the equation used to calculate the radius, we get the standard form of equation of a circle given by where is the center and is the radius of the circle. Equation of a Circle in Standard Form
- 14. 14 With Center at With Center at the Origin Equation of a Circle in Standard Form
- 15. 15 Equation of a Circle in Standard Form 𝒙𝟐 + 𝒚𝟐 =𝟏𝟔
- 16. 16 Equation of a Circle in Standard Form 𝒙𝟐 + 𝒚𝟐 =𝟗
- 17. 17 Equation of a Circle in Standard Form 𝒙𝟐 + 𝒚𝟐 =𝟒
- 18. 18 Equation of a Circle in Standard Form 𝒙𝟐 + 𝒚𝟐 =𝟏
- 19. 19 Equation of a Circle in Standard Form 𝒙𝟐 + 𝒚𝟐 =𝟎.𝟐𝟓
- 20. 20 What do you think will happen to the graph of a circle if ?
- 21. 21 If , then the graph is a single point (not a circle). Equation of a Circle in Standard Form
- 22. 22 What do you think will happen to the graph of a circle if ?
- 23. 23 If , then there is no graph since is imaginary. Equation of a Circle in Standard Form
- 24. Let’s Practice! 24 Find the equation of the circle with center at the origin and a radius of 10 units.
- 25. Let’s Practice! 25 Find the equation of the circle with center at and a radius of units.
- 26. Try It! 26 26 Find the equation of the circle with center at the origin and a radius of 12 units.
- 27. Let’s Practice! 27 Find the equation of the circle with center at and a radius of units.
- 28. Let’s Practice! 28 Find the equation of the circle with center at and a radius of units.
- 29. Try It! 29 29 Find the equation of the circle with center at and a radius of units.
- 30. 30 1. Solve for by equating to its corresponding binomial in the given equation. Finding the Center and Radius of a Circle Given Its Equation
- 31. 31 1. Solve for by equating to its corresponding binomial in the given equation. 2. Solve for by equating to its corresponding binomial in the given equation. Finding the Center and Radius of a Circle Given Its Equation
- 32. 32 1. Solve for by equating to its corresponding binomial in the given equation. 2. Solve for by equating to its corresponding binomial in the given equation. 3. Solve for by equating to its corresponding constant in the given equation. Finding the Center and Radius of a Circle Given Its Equation
- 33. Let’s Practice! 33 Find the center and the radius of the circle whose equation is
- 34. Let’s Practice! 34 Find the center and the radius of the circle whose equation is The center of the circle is at , and its radius measures units.
- 35. Try It! 35 35 Find the center and radius of the circle whose equation is
- 36. Tip 36 To identify the center of the circle given by the equation , we can simply get the additive inverse of and . Therefore, the center of the circle is at .
- 37. 37 When the standard form of equation of a circle is expanded, and the terms are arranged in decreasing order of powers, we get the general form of equation of a circle given by where , , and and are not zero at the same time. Equation of a Circle in General Form
- 38. Let’s Practice! 38 Identify the center and the radius of the circle defined by the equation .
- 39. Let’s Practice! 39 Identify the center and the radius of the circle defined by the equation . The center is at , and the radius is .
- 40. Try It! 40 40 Identify the center and radius of the circle defined by the equation .
- 41. Let’s Practice! 41 Find the general form of the circle illustrated below.
- 42. Let’s Practice! 42 Find the general form of the circle illustrated below.
- 43. Try It! 43 43 Find the general form of the circle illustrated below.
- 44. Let’s Practice! 44 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.
- 45. Let’s Practice! 45 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.
- 46. Let’s Practice! 46 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. Rowell is 31.62 feet away from his house. This is less than the radius of the circle. Thus, Rowell can still connect to their Wi-Fi at home.
- 47. Try It! 47 47 A cellular network company uses towers to transmit communication information. A tower located at 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.
- 48. Check Your Understanding 48 Fill in the table below by finding the standard form and the general form of the equation of the circle given the following data. Given Data Standard Form General Form 1. center at the origin with a radius of 9 cm 2. center at with a radius of cm
- 49. Check Your Understanding 49 Find the center and the radius of the circle defined by each equation. 1. 2. 3.
- 50. Check Your Understanding 50 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 -axis and that the ground is the - 𝑦 𝑥 axis.
- 51. Let’s Sum It Up! 51 ● A circle is formed when a plane perpendicular to the axis intersects a double-napped cone. ● A circle is the set of all points that are equidistant from a given point in the plane, called the center. ● Any segment with endpoints at the center and a point on the circle is a radius of the circle.
- 52. Key Formulas 52 Concept Formula Description Equation of a Circle in Standard Form where is the center of the circle is its radius Use this formula when finding the equation of a circle given its center and radius.
- 53. Key Formulas 53 Concept Formula Description Equation of a Circle in General Form This is the form of the equation when the standard form is expanded.
- 54. Challenge Yourself 54 54 In the definition of a circle, explain why the phrase “in a plane” is explicitly stated. If this phrase is not included, what geometric figure will be formed?
- 55. Photo Credits Bibliography 55 Slide 2: Sky Ranch, by Miki Mijares is licensed under CC BY-SA 3.0 via Wikimedia Commons. Barnett, Raymond, Michael Ziegler, Karl Byleen, and David Sobecki. College Algebra with Trigonometry. Boston: McGraw Hill Higher Education, 2008. Bittinger, Marvin L., Judith A. Beecher, David J. Ellenbogen, and Judith A. Penna. Algebra and Trigonometry: Graphs and Models. 4th ed. Boston: Pearson/Addison Wesley, 2009. Blitzer, Robert. Algebra and Trigonometry. 3rd ed. Upper Saddle River, New Jersey: Pearson/Prentice Hal, 2007. Larson, Ron. College Algebra with Applications for Business and the Life Sciences. Boston: MA:Houghton Mifflin, 2009. Simmons, George F. Calculus with Analytic Geometry. 2nd ed. New York: McGraw-Hill, 1996.