Analytic geometry lecture2
- 1. Analytic Geometry Lecture 2:Circles Engr. Adriano Mercedes H. Cano Jr. University of Mindanao College of Engineering Education Electronics Engineering MATH 201 1
- 2. Lecture Objectives Upon completion of this chapter, you should be able to: Learn basic concepts about Circle Plot a circle base on equation given. Solve problems involving equation of a circle 2
- 3. Outline Introduction Equation of a circle Graphing a circle Writing equation of a circle Forms of equation of a circle Terminologies Techniques in solving problems involving circle 3
- 4. Circle A set of points (locus) which are equidistant from a fixed point called center. The distance from the center to any points is called radius Two quantities that are needed to find the equation of a circle: Center Radius
- 5. Equation of a Circle centered at (0,0)
- 6. Example 1 Determine the center and radius of the given equation of circle.
- 7. Example 2 Consider the circle below. Fine the equation of the circle The center is at the origin To determine the radius: This leads to the equation of the circle:
- 8. Exercises Find the equation of the following circles with center at the origin and: radius √ 3 units passing through the point (–5 ; 12) passing through ( 1/2 ; 1/2 ) 8
- 9. Equation of a Circle centered at (h,k)
- 10. Example 3
- 11. Write the standard equation of the circle: with center at (4, 7) and radius of 5 units (x – 4)2 + (y – 7)2 = 25 Example 4
- 12. Example 5 Determine its center and radius. by completing the square and factorize Solution.
- 13. Write the standard equation of the circle: Center (2, -9) Radius of (x – 2)2 + (y + 9)2 = 11 11 Example 6
- 14. Exercise Determine the co-ordinates of the centre of the circle and the radius for each of the following: (x – 3)2 + (y – 2)2 = 9 x2 + y2 – x – 2y – 5 = 0 x2 + y2 + 2x – 6y + 9 = 0 Write the standard equation of the circle with center at (-3, 8) and a radius of 6.2 units
- 15. Exercise
- 16. General Equation of a Circle
- 17. SUMMARY x2 + y2 = r2 (x – h)2 + (y – k)2 = r2 Circle Standard Form General Equations Form
- 18. Terminologies 18
- 19. Terminologies 19
- 20. Terminologies 20 • A circumscribed circle or circumcircle passes through all vertices of a plane figure and contains the entire figure in its interior. • The center of this circle is called the circumcenter.
- 21. Notes!!! 21 For a polygon, each side of the polygon must be tangent to the circle. All triangles and regular polygons have circumscribed and inscribed circles. The radius of the circle is always perpendicular to the tangent line
- 22. Terminologies 22 • An inscribed circle is the largest possible circle that can be drawn on the inside of a plane figure. • For a regular polygon, the inradius (the radius of the inscribed circle) is called the apothem. • A unique circle inscribed to a triangle is called the incircle. For a polygon, each side of the polygon must be tangent to the circle. All triangles and regular polygons have circumscribed and inscribed circles.
- 23. Problems involving Equation of a circle Hints. Make a rough sketch of the problem Analyze Practice
- 24. 24
- 25. Points inside, on, outside a circle Two Methods 1. Calculate the distance from the center and compare this distance with the radius 2. Substitute the coordinates into the equations of the circle.
- 26. Example 7 26
- 27. Excersice 27
- 28. Intersection of a line and a circle 28
- 29. Example 8 29
- 30. Example 8 30
- 31. Example 9 31
- 32. Example 9 32
- 33. Exercise 33
- 34. Finding the equation of a circle: Given 3 points 34
- 35. Example 10 35
- 36. Exersice 36
- 37. Finding the equation of a circle: Given 2 points and equation of the line 37
- 38. Example 11 38
- 39. Example 11 39
- 40. Cramer’s rule 40
- 41. Cramer’s rule 41
- 42. Cramer’s rule 42
- 43. Cramer’s rule 43
- 44. Finding the equation of a circle: Given 2 points and equation of the tangent at one these points 44
- 45. Example 12 45 Slope of the perpendicular line: - ½ Equation of the perpendicular line: y – (-2) = (-1/2) (x-(-3)) y+2 = (-1/2) (x+3) 2y +4 = -x-3 x+y+7=0 g+f+7=0 eq1
- 46. Example 12 46 (-3,-2) on the circle: (3)^2 +(2)^2 +2g(-3)+2f(-2)+c=0 9+4-6g-4f+c=0 -6g-4f+c=0 eq 2 (0,-1) on the circle: (0) +(-1) +2g(0)+2f(-1)+c=0 -2f-1+c=0 eq 3
- 47. Exercise 47
- 48. Finding the equation of a circle: Given a radius, a point and equation of the line 48
- 49. Example 13 49
- 50. Example 13 50
- 51. Exercise 51
- 52. Equation of a tangent to a circle at a given point. 52
- 53. Example 14 53
- 54. Proving that a line is a tangent to a circle 54
- 55. Exercise 15 55
- 56. Length of a tangent to a circle from a point outside the circle 56
- 57. Exercise 16 57
- 58. Exercise 58
- 59. Exercise 59
- 60. Tangents parallel or perpendicular to a given line 60
- 61. Example 17 61
- 62. Equations from a tangent outside a circle 62
- 63. Example 18 63
- 64. Example 18 64