Precal Lesson 2 Circles lesson in mathematics

Precal Lesson 2 circles.pptx

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. 3 Define a circle (STEM-PC11AG-1a-2). Determine the standard form of equation of a circle (STEM-PC11AG-1a-3).

4. 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    

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 With Center at the Origin Equation of a Circle in Standard Form

15. 15 Equation of a Circle in Standard Form  

16. 16 Equation of a Circle in Standard Form  

17. 17 Equation of a Circle in Standard Form  

18. 18 Equation of a Circle in Standard Form  

19. 19 Equation of a Circle in Standard Form  

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. 24 Find the equation of the circle with center at the origin and a radius of 10 units.

25. 25 Find the equation of the circle with center at and a radius of units.    

26. 26 26 Find the equation of the circle with center at the origin and a radius of 12 units.

27. 27 Find the equation of the circle with center at and a radius of units.  

28. 28 Find the equation of the circle with center at and a radius of units.    

29. 29 29 Find the equation of the circle with center at and a radius of units.  

30. 30 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 Solve for by equating to its corresponding binomial in the given equation. 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 Solve for by equating to its corresponding binomial in the given equation. Solve for by equating to its corresponding binomial in the given equation. Solve for by equating to its corresponding constant in the given equation.   Finding the Center and Radius of a Circle Given Its Equation

33. 33 Find the center and the radius of the circle whose equation is  

34. 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. 35 35 Find the center and radius of the circle whose equation is  

36. 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. 38 Identify the center and the radius of the circle defined by the equation .  

39. 39 Identify the center and the radius of the circle defined by the equation .   The center is at , and the radius is .  

40. 40 40 Identify the center and radius of the circle defined by the equation .  

41. 41 Find the general form of the circle illustrated below.

42. 42 Find the general form of the circle illustrated below.  

43. 43 43 Find the general form of the circle illustrated below.

44. 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.   Find the equation of the circle in general form which describes the boundary of the Wi-Fi signal. Determine whether he can still connect to their Wi-Fi at home.

45. 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.   Find the equation of the circle in general form which describes the boundary of the Wi-Fi signal. Determine whether he can still connect to their Wi-Fi at home.  

46. 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.   Find the equation of the circle in general form which describes the boundary of the Wi-Fi signal. 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. 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. 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 Given Data Standard Form General Form 1. center at the origin with a radius of 9 cm

49. 49 Find the center and the radius of the circle defined by each equation. 1. 2. 3.  

50. 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 𝑦-axis and that the ground is the 𝑥-axis.

51. 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. 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. Concept Formula Description Equation of a Circle in Standard Form Use this formula when finding the equation of a circle given its center and radius.

53. 53 Concept Formula Description Equation of a Circle in General Form This is the form of the equation when the standard form is expanded. Concept Formula Description Equation of a Circle in General Form This is the form of the equation when the standard form is expanded.

54. 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. 55