3. Instructors: Dr. Lee Fong Lok Dr. Leung Chi Hong Student Name: Chan Mei Shan Student ID:S98039740 Topic:Coordinate Geometry (F3 Mathematics)

4. Distance Formula Slope Straight Line Drawing

5. Review:Distance and Slope Equation of Straight Lines Points of Division Perpendicular and Parallel Lines Intersection of Two Straight Lines

6. Equation of Special Lines Two Point Form Point-Slope Form Slope-Intercept Form Intercept Form General Form Back =>

7. Topic: Distance and Slope (Review) Back=>

8. P (x 1, y 1 ) Q (x 2,y 2 ) y 1 – y 2 distance ? By Pythagoras Theorem, x 1 – x 2 y1y1 y2y2 x2x2 x1x1 Distance

9. Example A = (-4, 2) B=(2, -4) (x 1,y 1 ) (x 2,y 2 )

10. Class Work (a) Find AB if A=(4,0) and B=(9,a) (Give the answer in terms of a.) (b) If AB=, find a. (a) (b) 2 min

11. P (x 1, y 1 ) Q (x 2,y 2 ) x 1 – x 2 y 1 – y 2 slope ? Slope

12. (4, 5) (1, 1) Example (x 2,y 2 ) (x 1,y 1 )

13. B(a 2, ab) A(b 2, –ab) Class Work Slope of AB? 3 min

14. (-4, 2) (2,2) If a line//x-axis slope = 0 Example

15. (2,-3) (2,2) If a line // y-axis slope is undefined Example zero!

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17. Topic: Point of Division Back=>

18. (9-y) (y -2) (8-x) (x-1) A= (1,2), B = (8,9) and AC : CB = 3 : 4 Find C(x,y) 4567821-2 5 6 7 8 9 2 1 x y 3 3 4 Point C? A(1,2) B(8,9) C(x,y) 4 3

19. B(8,9) (9-y) (8-x) 4 C(x,y) D (y -2) (x-1) A(1,2) C(x,y) 3 E ∵ ΔBCD ～ ΔCAE

20. B(8,9) A(1,2) C(x,y) 4 3 Observation x = 3 x 8 + 4 x 1 3 + 4 Calculation y = 3 x 9 + 4 x 2 3 + 4 Section Formula

21. A (1, 2) B (4, 8) P (a, b) 1 2 What are the coordinates of P ? Ans: P = (2, 4) Example

22. A (a, b) B (4, 9) P (3, 1) 5 2 Find the values of a and b ClassWork 5 min

23. Solution

25. A (3, -7) B (5, 3) P (a, b) 1 1 Find the coordinates of point P Class Work 3 min

26. A(x 1,y 1 ) B (x 2,y 2 ) P (a, b) 1 1 P is the mid-point of AB

27. 1 2 1 2 1 2 B (5, -2) C (-2, -5) A (3, 4) P G (4, 1) Let P = (a, b) & G = (p, q) Example

28. B (x2,y2) C (x3, y3) A (x1,y1) G(x,y) Observation Given : G is the centroid of △ ABC Then:

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30. A (-3, 4) B (2, 1) P Find the ratio of AP : PB. Challenge Answer =>

31. Let AP : PB = 1 : k Solution Back =>

32. Topic: Equations of Special Lines Back=>

33. (1, 3) (3, 3) (-3, 3) (-1, -3) (2, -3) (-5, -3) x = -3 y = 3 y = -3 Horizontal Lines (2,1) y = 1

34. Vertical Lines (2, 2) (2, 0) (-3, -3) x = -3 x = 2 x = -1 x = -3 (-1, -4)

35. (a, b) L2L2L2L2 L1L1L1L1 P Ans: 1. L 1 : x = a L 2 : y = b 2. P=(0, b) Class Work 0 x yFind The equations of L 1 and L 2;The equations of L 1 and L 2; The coordinates of point P.The coordinates of point P. 3 min

36. (1,1) (3,3) (-3,-3) y = x (1,-2) (-2,4) (-1,2) y =-2x Straight lines Passing through origin Straight lines Passing through origin

37. (a,b) Observation

38. Class Work (6,7) x y L2L2L2L2 L1L1L1L1 (4,-3) Ans: L 1 : L 1 : L 2 : L 2 : 3 min Find the equations of L 1 and L 2.

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40. Topic: Two-Point Form Back=>

41. Find the equation of L. B(5, 8) A(1, 3) P(x, y)L M AP = M AB

42. B(5, 8) A(1, 3) P(x, y) L: 5x-4y+7=0 M BP = M AB Will the result be the same if We consider M BP instead of M AP ?

43. (-4, 4) (2, -3) L (-2, b) P (a) Find the equation of L. (b)Find the value of b. (c)Find the coordinates of P. L: 7x + 6y + 4 = 0 Example

44. (a)Find the equation of the straight line joining (-3, 2) and (2, -1). (b)Does the point (7, -4) lie on the straight line ? (c)State whether the point (3, -2) lies on the straight line or not. The point (7, -4) lies on the straight line. The point (3, -2) does not lie on the line. Example L: 3x - 5y + 1 = 0

45. Class Work (a)Find the equation of the straight line which passes through (0,0) and (-4,-6). (b)If the point A(a,3) lies on L, find a. Solution. (a)(b) 7 min

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47. Topic: Point-Slope Form Back=>

48. Point-slope Form A(1, 7) L B(x, y) slope = 3 M AB = Slope

49. Find the equation of the line which passes through (-1,-5) and has slope -3 : Example Working? Solution

50. (a) Find the equation of L. (b)What is the value of b ? Put B(2, b) into the equation L: x + 3y - 3 = 0 Example (-3, 2)L B (2, b)

51. Find (a) The equation of L. (b) The coordinates of P (b) The coordinates of P (c) The coordinates of Q (c) The coordinates of QL P (a, 2) Q (-2, 0) Class Work 10 min

52. Solution. (a) Put P(a, 2) into L therefore P = (-3.5, 2) (b) Let Q = (0, b) Q = (0, ) (c)

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54. Topic: Slope-Intercept Form

55. y-intercept y-intercept x-intercept x-intercept L1L1L1L1 (0, 3) (-2, 0) L 1 cuts the y-axis at point (0,3) L 1 cuts the x-axis at point (-2,0) Intercepts -3-212345 -2 1 2 3 4 5 6 -3 x y -4 0

56. What is the equation of L ? slope slope y-intercept y-intercept (x, y)L (0, 4) 4 slope = 3 Slope-intercept Form

57. Example (a)Find the equation of the straight line with y-intercept –1 and slope –3 in the slope-intercept form. (b)What is the slope and the y-intercept of the straight line y = 3x - 7 ? y=  3x  1 (c)What is the slope and the y-intercept of the straight line 2x + 3y – 1 = 0 ? Slope-intercept Form

58. Example L : kx + 3y – 2k = 0 with slope –2. L : kx + 3y – 2k = 0 with slope –2. (a) Find the value of k. (a) Find the value of k.thus, (b)What is the y-intercept of L ? Slope-intercept Form

59. 7 min Slope-intercept Form Class Work Find the value of k for the following straight line, L. L : 3x + 4y + k = 0 with y-intercept 5. thus, Alternatively, Put (0, 5) into the equation of L. Ans.

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61. Topic: Intercept Form Back=>

62. 2 3 B(2, 0) A(0, 3) L y-interceptx-intercept thus, P(x, y) M AP = M AB What is the equation of L ?

63. Find the equation of L in intercept form. Example Ans: Do the point (4, 6) and (12, 9) lie on L ? x-intercept -4 y-intercept 3 L The point (4, 6) lies on L. Put (4, 6) into the equation Put (12, 9) into the equation (12, 9) does not lie on L.

64. (a)Convert 7x + 4y + 28 = 0 into the intercept form. (b)What are the x-intercept and y-intercept of the straight line ? x-intercept = -4 and y-intercept = -7 Example

65. Find the area of the shaded region. The area of the shaded region is Example x y 0 L : 2x+ 5y + 10 = 0 Intercept form -2 -5

66. Class Work (a) Find the intercepts of L 1. (b) Find the equation of L 2. (c) Find the area of the shaded region. x y 0 L 1: 3x + 5y-15=0 L2L2L2L2 -2 10 min

67. Solution. x-intercept = 5 and y-intercept = 3. (a) The equation of L 2 is (b) The area of the shaded region is (c) x y 0 L 1: 3x + 5y-15=0 L2L2L2L2 -2 5 3

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69. Topic: General Form Back=>

70. Ax + By + C = 0 Convert into the general form. General Form

71. Class Work Convert into the general form.

72. What are the slope and the y-intercept of the straight line 4x – 3y + 7 = 0 ? Example

73. Find the equation of L in the general form. x y 0 -7 slope = -2 L Example

74. Find the x-intercept and the y-intercept of the straight line 12x – 7y + 4 = 0. Example

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76. Topic: Parallel Lines and Perpendicular Lines Back=>

77. If L 1 // L 2, then m L1 = m L2 What will happen if Two lines L 1 and L 2 Are parallel? A FACT to know... Conversely, if m L1 = m L2 Then L 1 // L 2

78. Example L1L1L1L1 L2L2L2L2 slope = 2 Determine whether L 1 // L 2 Since m 1 = m 2 = 2, then, L 1 is parallel to L 2

79. Find the equation of L 2 xy 0 (0, 0) L 1 : m = 2 L2L2L2L2 Example m L2 = m L1 = 2 By point-slope form,

80. 4 min (a) Find the equation of L 2. -5 x y L 1 : slope = -3 0 (-5, 0) L2L2L2L2 (b) Does the point (-3, -5) lies on L 2 ? L.H.S. = = 3(-3) + (-5) + 15 = 1  R.H.S. Thus, (-3, -5) does not lie on L 2 Class Work More...

81. Find the equation of L 2. x y 0 L 1 : 4x + 3y + 12 = 0 L2L2L2L2 -6 (-6, 0) Express L 1 into slope intercept form. Step 1: Express L 1 into slope intercept form. Find the slope of L 2 Step 2: Find the slope of L 2 m L2 = m L1 = Use point-slope form to find L 2. Step 3: Use point-slope form to find L 2. Example

82. 6 min Steps : 1.Express the given line into slope-intercept form. 2.Find the slope of L1. 3. Use point-slope form to find the equation of the line. Find the equation of the line L1 which is parallel to 3x + 2y – 5 = 0 and passes through (4, -1). Class Work

83. Solution. Step 1: Express the given line into slope-intercept form. Step 2: Find m L. m L = Step 3: Use point-slope form to find the equation

84. If L 1 ⊥ L 2, then m L1 x m L2 =-1 One more FACT... Conversely, if m L1 x m L2 =-1 Then L 1 ⊥ L 2

85. Find the coordinates of P.(Hint: Let P = (a,0) thus, P = (-0.5, 0) L1L1L1L1 x y 0 L2L2L2L2 slope = 2 P Example ∵ L 1 ⊥ L 2 ∴ m L1 x m L2 =-1

86. Find the equation of L 2. Express L 1 into slope intercept form. Step 1: Express L 1 into slope intercept form. Find the slope of L 2 Step 2: Find the slope of L 2 m L2 = -1÷m L1 =-1÷ Use point-slope form to find L 2. Step 3: Use point-slope form to find L 2. Example x y 0 (-2, -3) L 1 : 3x-2y +10 =0 L2L2L2L2

87. 6 min Steps : 1.Express the given line into slope-intercept form. 2.Find the slope of L. 3. Use point-slope form to find the equation of the line. Find the equation of the line L which is perpendicular to 3x - 2y + 6 = 0 and pases through (-4, 3). Class Work

88. Solution. Step 1: Express the given line into slope-intercept form. Step 2: Find m L. m L = Step 3: Use point-slope form to find the equation

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91. Find the equation of the perpendicular bisector of the line segment joining (3, -5) and (-7, 9). [ Ans.: 5x - 7y + 24 = 0 ] Steps : 1.Find the coordinates of the midpoint. 2.Find the slope of the line segment. 3.Find the slope of the perpendicular bisector 4.Use point-slope form to find the equation of the line. Challenge Back=>

92. Topic: Point of Intersection Back=>

93. x y 0 y = 7 x = 5 P What are the coordinates of P ? A. P = (-5, -7) A. P = (-5, -7) B. P = (-5, 7) B. P = (-5, 7) C. P = (5, -7) C. P = (5, -7) D. P = (5, 7) D. P = (5, 7) E. P = (7, 5) E. P = (7, 5)

94.  You are wrong !  Don’t give up …  Try it again … return Sorr y !

95. Correct ! Return

96. What are the coordinates of P ? A. P = (-5, 7) A. P = (-5, 7) B. P = (5, 7) B. P = (5, 7) C. P = (7, 2) C. P = (7, 2) D. P = (7, 13) D. P = (7, 13) E. P = (13, 7) E. P = (13, 7) x y 0 x = 36 P y = 3x – 8

97. What are the coordinates of P ? x y 0 2x – 5y + 16 = 0 3x + 2y + 5 = 0 P P = (-3, 2) Example

98. Find the coordinates of the point of intersection of The coordinates are (5, 4) Example

99. P = (1, 2) What are the coordinates of P ? y 0 3x – 4y + 5 = 0 P 2 4 L L : Example

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