Vardaan
Class 9 Maths • Chapter 3 • NCERT + RS Aggarwal + RD Sharma

Coordinate Geometry

Vardaan Learning Institute  |  School-Exam Focused Notes

📍 1. The Cartesian Plane

X Y O 1 2 3 −1 −2 1 2 −1 I (+,+) II (−,+) III (−,−) IV (+,−) A(3,2) B(−2,2) C(−1,−2) D(3,−2) X' Y'
Key Vocabulary:
• Cartesian Plane / Coordinate Plane: 2D plane formed by two perpendicular lines
• X-axis: Horizontal number line (goes right: positive, left: negative)
• Y-axis: Vertical number line (goes up: positive, down: negative)
• Origin (O): Intersection of axes, coordinate = (0, 0)
• Ordered Pair (x, y): x = abscissa (x-coordinate), y = ordinate (y-coordinate)
• Abscissa: Distance from Y-axis (perpendicular distance to Y-axis)
• Ordinate: Distance from X-axis (perpendicular distance to X-axis)

📊 2. The Four Quadrants

Quadrant Sign of x Sign of y Example Point Location
I (First) + (positive) + (positive) (3, 4) Top-Right
II (Second) − (negative) + (positive) (−2, 5) Top-Left
III (Third) − (negative) − (negative) (−3, −1) Bottom-Left
IV (Fourth) + (positive) − (negative) (4, −3) Bottom-Right
On X-axis any 0 (5, 0), (−2, 0) y = 0
On Y-axis 0 any (0, 3), (0, −4) x = 0

🔎 3. Plotting and Reading Points

How to Plot a Point P(x, y):
1. Start at the Origin O.
2. Move |x| units right (if x positive) or left (if x negative) along X-axis.
3. From that position, move |y| units up (if y positive) or down (if y negative).
4. Mark and label the point.

Example: Plot P(−3, 4):
Move 3 units LEFT on X-axis, then 4 units UP → Point is in Quadrant II ✓

🏗️ 4. Key Facts for Exam

Statement Explanation
Coordinates of origin (0, 0)
A point on X-axis Its y-coordinate = 0. Form: (a, 0)
A point on Y-axis Its x-coordinate = 0. Form: (0, b)
Mirror image in X-axis (x, y) → (x, −y)
Mirror image in Y-axis (x, y) → (−x, y)
Mirror image in Origin (x, y) → (−x, −y)
Point equidistant from both axes |x| = |y|, i.e., on lines y = x or y = −x
Three collinear points Area of triangle formed by them = 0
Distance Formula (Preview — taught in Class 10 but useful here):
Distance between A(x₁, y₁) and B(x₂, y₂) = √[(x₂−x₁)² + (y₂−y₁)²]

Area of Triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃):
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
(Collinear if area = 0)
✏️ Practice Questions — Coordinate Geometry (28 Questions)

Section A — 1 Mark Easy

Q1. Write the coordinates of the origin.
Q2. In which quadrant does (−3, +4) lie?
Q3. A point has abscissa 0. Where does it lie?
Q4. What is the ordinate of every point on the X-axis?
Q5. Name the axes and the plane in Cartesian geometry.
Q6. In which quadrant: (−5, −3)?
Q7. What are the signs of coordinates in Quadrant IV?
Q8. Plot A(2,3), B(−1,4), C(0,−5), D(−3,−2) on a coordinate plane. (Describe where each point falls.)

Section B — 2/3 Mark Medium

Q9. Find the mirror image of (3, −5) in (i) X-axis, (ii) Y-axis, (iii) Origin.
Q10. A point lies 3 units to the left of Y-axis and 4 units above X-axis. Write its coordinates.
Q11. Points P and Q are symmetric about the X-axis. If P = (4, −3), find Q.
Q12. Without plotting, determine the quadrant of (−1)(2), (−2)(−3), (3)(−1). (Numbers represent signs of x and y.)
Q13. On a graph, the y-coordinate of a point equals its x-coordinate. Give 4 points satisfying this condition.
Q14. A rectangle has vertices at (1,3), (4,3), (4,1), (1,1). Find its length and breadth from coordinates.
Q15. Verify: are the points (3,0), (0,4), (3,4) vertices of a right triangle? (Use coordinates.)
Q16. If a point (a, b) lies in Quadrant III, what are the signs of a and b? What about (−a, b)?

Section C — 3/4 Mark Medium

  1. Q17. Plot the vertices of a square with one vertex at origin, side = 5 units (all in Quadrant I). Name the vertices.
  2. Q18. A point A(a, b) satisfies 2a + 3b = 12. Name 3 different coordinates of A and plot them.
  3. Q19. Points A(1,1), B(4,1), C(3,4) form a triangle. Plot and find: which type of triangle (by observation of side lengths using distance formula)?
  4. Q20. What is the shape formed by (0,0), (4,0), (4,3), (0,3)? Name and find its area from coordinates.
  5. Q21. Describe all points that are equidistant from both coordinate axes.

Section D — Challenge Hard

  1. Q22. Show that the points (1,2), (3,4), (5,6) are collinear using the area formula.
  2. Q23. A point moves such that its ordinate is always twice its abscissa. Write 4 pairs of coordinates and name the shape traced.
  3. Q24. P(2,5) is reflected over Y-axis, then the image is reflected over X-axis. Find the final position.
  4. Q25. The vertices of a triangle are A(−2,0), B(2,0) and C(0,4). Find the area of this triangle.
  5. Q26. How many points lie on the X-axis whose distance from the origin is exactly 5? Name them.
  6. Q27. Find the perimeter of rectangle with vertices A(0,0), B(5,0), C(5,3), D(0,3).
  7. Q28. A point P(a, b) moves to P'(a+2, b−1). If P starts at (1, 4), where is P after 3 moves?
✅ Key Answers: Q1:(0,0) | Q2:II | Q3:Y-axis | Q4:0 | Q6:III | Q7:(+,−) | Q9:(3,5),(−3,−5),(−3,5) | Q10:(−3,4) | Q16:(−a,b) in II | Q22:Area=0, so collinear ✓ | Q25:Area=8 sq.units | Q26:2 points: (5,0) and (−5,0) | Q27:Perimeter=16 | Q28:(7,1) after 3 moves

📝 Quick Revision

  1. The Cartesian plane has X-axis (horizontal) and Y-axis (vertical) intersecting at Origin O(0,0).
  2. A point (x, y): x = abscissa (from Y-axis), y = ordinate (from X-axis)
  3. Quadrant I:(+,+) | II:(−,+) | III:(−,−) | IV:(+,−)
  4. On X-axis: y=0. On Y-axis: x=0.
  5. Mirror in X-axis:(x,y)→(x,−y) | in Y-axis:(x,y)→(−x,y) | in Origin:(x,y)→(−x,−y)
  6. Collinear points: Area of triangle formed = 0