Chapter 4 — Coordinate Geometry

1. Cartesian Plane

Two perpendicular axes form the coordinate plane:

• x-axis
• y-axis

Their intersection is the origin:

(0,0)

A point is represented by:

(x,y)

where:

• x = abscissa
• y = ordinate

2. Four Quadrants

Quadrant	Sign of x	Sign of y
I	+	+
II	−	+
III	−	−
IV	+	−

A point lying on an axis is not in any quadrant.

3. Reflection of Points

Reflection in the x-axis:

(x,y) → (x,−y)

Reflection in the y-axis:

(x,y) → (−x,y)

Reflection in the origin:

(x,y) → (−x,−y)

The key idea is to identify which coordinate changes sign.

4. Distance Between Two Points

For:

A(x₁,y₁) and B(x₂,y₂)

the distance is:

AB = √[(x₂−x₁)² + (y₂−y₁)²]

This comes directly from the Pythagorean relationship.

5. Midpoint

The midpoint of two points is obtained by averaging their corresponding coordinates:

M = ((x₁+x₂)/2, (y₁+y₂)/2)

6. Slope

The slope measures the steepness of a line.

For two points:

m = (y₂−y₁)/(x₂−x₁)

A horizontal line has slope 0.

A vertical line has an undefined slope.

7. Intercepts

x-intercept: point where the line meets the x-axis, so y = 0.

y-intercept: point where the line meets the y-axis, so x = 0.

8. Equations of a Straight Line

The chapter presents three useful forms:

General form

Ax + By + C = 0

Slope-intercept form

y = mx + c

where m is slope and c is the y-intercept.

Intercept form

x/a + y/b = 1

where a and b are the x- and y-intercepts respectively.

Exam Focus

Practise moving confidently between:
points → distance → midpoint → slope → equation of line.

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Chapter 4 — Coordinate Geometry: Complete Question Bank

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A. Multiple Choice Questions (MCQs)

1. The point where the x-axis and y-axis intersect is called the:

A. Quadrant
B. Origin
C. Intercept
D. Vertex

2. In the point (−4, 7), the abscissa is:

A. 7
B. −7
C. −4
D. 4

3. In which quadrant is the point (−3, −5) located?

A. I
B. II
C. III
D. IV

4. The signs of coordinates of a point in Quadrant IV are:

A. (+,+)
B. (−,+)
C. (−,−)
D. (+ ,−)

5. The reflection of (5, −2) in the y-axis is:

A. (−5, −2)
B. (5, 2)
C. (−5, 2)
D. (2, −5)

6. The reflection of (−3, 8) in the x-axis is:

A. (3,8)
B. (−3,−8)
C. (3,−8)
D. (−8,−3)

7. The reflection of (4, −6) in the origin is:

A. (−4,6)
B. (4,6)
C. (−4,−6)
D. (6,−4)

8. The perpendicular distance of (x,y) from the y-axis is:

A. |y|
B. x+y
C. |x|
D. x−y

9. The perpendicular distance of (x,y) from the x-axis is:

A. |x|
B. |y|
C. x+y
D. xy

10. The slope of a horizontal line is:

A. 1
B. −1
C. 0
D. Undefined

11. The slope of a vertical line is:

A. 0
B. 1
C. −1
D. Undefined

12. The slope of the line through (x₁,y₁) and (x₂,y₂) is:

A. (x₂−x₁)/(y₂−y₁)
B. (y₂−y₁)/(x₂−x₁)
C. (x₁+x₂)/(y₁+y₂)
D. (y₁−y₂)/(x₁+x₂)

13. Two non-vertical lines are parallel when their slopes are:

A. Opposite
B. Equal
C. Reciprocal
D. Zero

14. Two non-vertical lines are perpendicular when the product of their slopes is:

A. 0
B. 1
C. −1
D. 2

15. The y-intercept of a line always has the form:

A. (x,0)
B. (0,y)
C. (x,x)
D. (y,0)

16. The x-intercept is obtained algebraically by putting:

A. x=0
B. y=0
C. x=y
D. x=1

17. Which is the slope-intercept form of a straight line?

A. Ax+By+C=0
B. y=mx+c
C. x/a+y/b=1
D. xy=c

18. In y=mx+c, c represents the:

A. Slope
B. x-coordinate
C. y-intercept
D. x-intercept

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B. Fill in the Blanks

19. The horizontal axis of the Cartesian plane is called the __________.

20. The vertical axis is called the __________.

21. The point (0,0) is called the __________.

22. The x-coordinate of a point is called its __________.

23. The y-coordinate is called its __________.

24. Quadrant I has signs __________.

25. Quadrant II has signs __________.

26. Quadrant III has signs __________.

27. Quadrant IV has signs __________.

28. Reflection in the y-axis changes the sign of the __________ coordinate.

29. Reflection in the x-axis changes the sign of the __________ coordinate.

30. Reflection in the origin changes the signs of __________ coordinates.

31. The slope of a line measures its __________ and direction.

32. The slope of a line is commonly described as __________ over run.

33. A horizontal line has __________ slope.

34. A vertical line has __________ slope.

35. In y=mx+c, m represents the __________.

36. In y=mx+c, c represents the __________.

37. The general equation of a straight line is __________.

38. The intercept form of a line is __________.

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C. True or False

39. The order of coordinates in a point is important.

40. (3,−2) and (−2,3) represent the same point.

41. A point on the x-axis has y-coordinate zero.

42. A point on the y-axis has x-coordinate zero.

43. The origin belongs to all four quadrants.

44. A point in Quadrant II has a negative x-coordinate and positive y-coordinate.

45. Reflection in the y-axis changes only the x-coordinate.

46. Reflection in the x-axis changes only the y-coordinate.

47. Reflection in the origin changes both coordinate signs.

48. Distance from an axis can be negative.

49. A positive slope means the line rises from left to right.

50. Parallel non-vertical lines have equal slopes.

51. Perpendicular non-vertical lines have equal slopes.

52. A vertical line has an undefined slope.

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D. Match the Following

Column A	Column B
53. Origin	a. y=mx+c
54. Slope-intercept form	b. (0,0)
55. x-intercept	c. (x,0)
56. y-intercept	d. (0,y)
57. Horizontal line	e. Slope 0
58. Vertical line	f. Undefined slope

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E. Identify the Quadrant

59. Identify the quadrant containing A(4,6).

60. Identify the quadrant containing B(−7,3).

61. Identify the quadrant containing C(−2,−9).

62. Identify the quadrant containing D(8,−4).

63. A point has a negative x-coordinate and positive y-coordinate. Which quadrant contains it?

64. A point has both coordinates negative. Which quadrant contains it?

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F. Reflections of Points

65. Find the reflection of (6,4) in the y-axis.

66. Find the reflection of (−7,2) in the y-axis.

67. Find the reflection of (5,−3) in the x-axis.

68. Find the reflection of (−4,−8) in the x-axis.

69. Find the reflection of (3,−9) in the origin.

70. A point (a,b) is reflected in the y-axis. Write its new coordinates.

71. A point (p,q) is reflected in the x-axis. Write its new coordinates.

72. A point (r,s) is reflected in the origin. Write its new coordinates.

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G. Coordinates as Perpendicular Distances

73. Find the distance of P(−8,5) from the y-axis.

74. Find the distance of Q(−3,−7) from the x-axis.

75. A point lies in Quadrant II and is 6 units from the x-axis and 9 units from the y-axis. Find its coordinates.

76. A point lies in Quadrant IV. Its distance from the y-axis is 5 units and from the x-axis is 8 units. Find its coordinates.

77. A point has coordinates (a,b) and lies in Quadrant III. State the signs of a and b.

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H. Slope

78. Find the slope of the line joining (1,2) and (5,10).

79. Find the slope of the line joining (−2,5) and (4,−7).

80. Find the slope of the line joining (3,6) and (8,6).

81. Find the slope of the line joining (4,2) and (4,9).

82. Determine whether the line joining (2,3) and (6,11) rises or falls from left to right.

83. What is the slope of a line parallel to the x-axis?

84. What can you say about the slope of a line parallel to the y-axis?

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I. Parallel and Perpendicular Lines

85. A line has slope 4. What is the slope of any non-vertical line parallel to it?

86. A line has slope 3/5. Find the slope of a line perpendicular to it.

87. Determine whether lines with slopes 2 and 1/2 are perpendicular.

88. Determine whether lines with slopes 5 and 5 are parallel.

89. A line has slope −3. Find the slope of a perpendicular line.

90. Two lines have slopes −2 and 1/2. What is their relationship?

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J. Intercepts

91. Find the x-intercept and y-intercept of:

y = 3x − 9

92. Find the x-intercept and y-intercept of:

y = −2x + 8

93. Find the y-intercept of:

4x + y − 12 = 0

94. Find the x-intercept of:

5x − 2y + 10 = 0

95. What coordinate must be zero at an x-intercept?

96. What coordinate must be zero at a y-intercept?

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K. Equation of a Straight Line

97. Write the slope-intercept equation of a line having slope 3 and y-intercept 5.

98. Write the equation of a line having slope −2 and y-intercept 7.

99. Find the equation of the line with slope 4 passing through (0,−3).

100. Find the equation of the line passing through (−2,5) and (2,−3).

101. Convert:

3x−2y+6=0

into slope-intercept form.

102. Convert:

5x+y−10=0

into slope-intercept form.

103. Write the equation of a line whose x-intercept is a and y-intercept is b.

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L. Assertion–Reason Questions

Choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

104. Assertion: The reflection of (3,5) in the y-axis is (−3,5).
Reason: Reflection in the y-axis changes the sign of the x-coordinate.

105. Assertion: A horizontal line has zero slope.
Reason: Its vertical change is zero.

106. Assertion: Two non-vertical lines with slopes 4 and 4 are parallel.
Reason: Parallel non-vertical lines have equal slopes.

107. Assertion: A vertical line has slope zero.
Reason: Its horizontal change is zero.

108. Assertion: The x-coordinate of an x-intercept is not necessarily zero.
Reason: Every point on the x-axis has y-coordinate zero.

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M. Higher-Order Thinking Questions

109. A point lies in Quadrant II. Its distance from the x-axis is 4 units and its distance from the y-axis is 7 units. Determine the point.

110. A point (a,b) is reflected first in the y-axis and then in the x-axis. What single transformation gives the same final point?

111. Two points A and B have the same y-coordinate but different x-coordinates. What type of line joins them? What is its slope?

112. Two points have the same x-coordinate but different y-coordinates. What type of line joins them? What is its slope?

113. A line has positive slope and crosses the positive y-axis. Describe its general direction on the coordinate plane.

114. The line 2x−3y+6=0 is parallel to another line. What must be true about the other line’s slope?

115. A line has slope 2. Another line passes through (1,3) and (5,−5). Determine whether the two lines are perpendicular, parallel, or neither.

116. A line passes through (2,5) and has slope −3. Find its equation.

117. A line passes through (−3,4) and (1,−4). Find its slope and determine whether it rises or falls from left to right.

118. A line has x-intercept 6 and y-intercept −3. Write its equation in intercept form.

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N. Application-Based Questions

119. A robot moves from (2,3) to (8,3) on a coordinate grid. What is the nature of its path and what is its slope?

120. A lift moves vertically along the line x=5. Explain why its slope is undefined.

121. A road on a coordinate map passes through (0,2) and (4,10). Find its slope and equation.

122. A designer reflects a point (−6,4) across the y-axis and then across the x-axis. Find the final position.

123. A map uses coordinates to represent two locations (−4,7) and (−4,−2). Are the locations joined by a horizontal or vertical line? Explain.

124. A straight path has equation y=3x−12. Find where it crosses each coordinate axis.

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Answer Key

MCQs

1. B
2. C
3. C
4. D
5. A
6. B
7. A
8. C
9. B
10. C
11. D
12. B
13. B
14. C
15. B
16. B
17. B
18. C

Fill in the Blanks

19. x-axis
20. y-axis
21. origin
22. abscissa
23. ordinate
24. (+,+)
25. (−,+)
26. (−,−)
27. (+ ,−)
28. x
29. y
30. both
31. steepness
32. rise
33. zero
34. undefined
35. slope
36. y-intercept
37. Ax+By+C=0
38. x/a + y/b = 1

True/False

39. True
40. False
41. True
42. True
43. False
44. True
45. True
46. True
47. True
48. False
49. True
50. True
51. False
52. True

Match

53–b
54–a
55–c
56–d
57–e
58–f

Numerical Answers

59. Quadrant I
60. Quadrant II
61. Quadrant III
62. Quadrant IV
63. Quadrant II
64. Quadrant III
65. (−6,4)
66. (7,2)
67. (5,3)
68. (−4,8)
69. (−3,9)
70. (−a,b)
71. (p,−q)
72. (−r,−s)
73. 8 units
74. 7 units
75. (−9,6)
76. (5,−8)
77. a<0, b<0
78. 2
79. −2
80. 0
81. Undefined
82. Rises
83. 0
84. Undefined
85. 4
86. −5/3
87. No
88. Yes
89. 1/3
90. Perpendicular
91. x-intercept (3,0), y-intercept (0,−9)
92. x-intercept (4,0), y-intercept (0,8)
93. (0,12)
94. (−2,0)
95. y-coordinate
96. x-coordinate
97. y=3x+5
98. y=−2x+7
99. y=4x−3
100. y=−2x+1
101. y=(3/2)x+3
102. y=−5x+10
103. x/a+y/b=1

Assertion–Reason

104. A
105. A
106. A
107. D
108. A

Higher-Order Answers

109. (−7,4)
110. Reflection in the origin
111. Horizontal line; slope = 0
112. Vertical line; slope is undefined
113. It rises from left to right and crosses the positive y-axis.
114. Its slope must be 2/3.
115. The second line has slope −2; since 2(−2)=−4, neither parallel nor perpendicular.
116. y=−3x+11
117. Slope = −2; it falls from left to right.
118. x/6 + y/(−3) = 1, equivalently x/6 − y/3 = 1.
119. Horizontal path; slope = 0
120. Because the change in x is zero, so the slope is undefined.
121. Slope = 2; equation = y=2x+2
122. First reflection: (6,4); second: (6,−4)
123. Vertical line, because both points have the same x-coordinate.
124. x-intercept = (4,0); y-intercept = (0,−12).