Introduction
Coordinate Geometry is one of the most imp and practical branches of mathematics that helps us locate any point precisely in a plane using two perpendicular lines. This chapter forms the foundation for understanding how positions and distances work in two-dimensional space.
What is Coordinate Geometry?
Coordinate Geometry is the branch of mathematics that uses coordinates (numbers) to describe the position of points in a plane. It was developed by the French mathematician René Descartes in the 17th century, which is why it’s also called the Cartesian System.
Real-Life Examples of Coordinate System
Example 1: Street with Houses Imagine a city with streets running in different directions. To find a specific house, you need TWO pieces of information:
- Which street (column number)
- What house number on that street (row number)
Example 2: Dot on Paper To locate a dot on a sheet of paper precisely:
- Distance from the left edge: 5 cm
- Distance from the bottom edge: 9 cm
These two measurements fix the position of the dot exactly!
Section 1: The Cartesian System
The Cartesian Coordinate System
The Cartesian Coordinate System
Y-axis (Vertical)
↑
|
6 |
5 |
4 | Q-II | Q-I
3 | (−,+) O| (+,+)
2 | |
1 | |
0 |__________|___________ X-axis (Horizontal)
-1 | |
-2 | |
-3 | Q-III | Q-IV
-4 | (−,−) | (+,−)
-5 |
-6 |
|
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6What are the Axes?
In a Cartesian system, we use two perpendicular lines to locate points:
1. The X-axis (Horizontal Line)
- Runs from left to right
- Positive numbers to the right of origin
- Negative numbers to the left of origin
2. The Y-axis (Vertical Line)
- Runs from bottom to top
- Positive numbers above the origin
- Negative numbers below the origin
3. The Origin (O)
- The point where both axes meet
- Coordinates of origin: (0, 0)
The Four Quadrants
The two axes divide the plane into FOUR equal parts called Quadrants:
- Quadrant I: (+, +) – Both positive
- Quadrant II: (−, +) – x negative, y positive
- Quadrant III: (−, −) – Both negative
- Quadrant IV: (+, −) – x positive, y negative
The quadrants are numbered anticlockwise starting from the right.
Section 2: Understanding Coordinates
What are Abscissa and Ordinate?
Every point in the coordinate plane is identified by TWO numbers called coordinates, written as (x, y):
Abscissa (x-coordinate):
- The perpendicular distance of a point from the y-axis
- Measured along the x-axis
- Positive if to the right of y-axis
- Negative if to the left of y-axis
Ordinate (y-coordinate):
- The perpendicular distance of a point from the x-axis
- Measured along the y-axis
- Positive if above x-axis
- Negative if below x-axis
Imp Rule About Coordinates
The order matters!
- (4, 3) and (3, 4) are NOT the same points
- Always write x-coordinate first, then y-coordinate
- (4, 3) means: 4 units right, 3 units up
Points on the Axes
Points on x-axis: Always have y-coordinate = 0
- Form: (x, 0)
- Examples: (2, 0), (−5, 0), (0, 0)
Points on y-axis: Always have x-coordinate = 0
- Form: (0, y)
- Examples: (0, 3), (0, −4), (0, 0)
The Origin: (0, 0)
- Zero distance from both axes
Section 3: Locating Points on the Coordinate Plane
Step-by-Step Process:
Locating a Point P at Coordinates (4, 3)
Y-axis
↑
6 |
5 |
4 |
3 |...........P(4, 3)
2 |
1 |
0 |_________________ X-axis
-1 |
-2 |
-3 |
|
0 1 2 3 4 5 6
Steps to plot (4, 3):
1. Start at Origin O(0, 0)
2. Move 4 units RIGHT along X-axis
3. Move 3 units UP along Y-axis
4. Mark point PTo plot a point like (4, 3):
- Start at the origin (0, 0)
- Move 4 units to the right (along x-axis) since x = 4
- From there, move 3 units upward (along y-axis) since y = 3
- Mark the point where you stop
To read a point:
- Draw a perpendicular from the point to the x-axis → Read x-coordinate
- Draw a perpendicular from the point to the y-axis → Read y-coordinate
Section 4: Solved Examples
Example 1: Reading Coordinates from a Diagram
Given Figure: Points B, M, L, S are marked on a coordinate plane.
| Point | x-coordinate | y-coordinate | Coordinates |
|---|---|---|---|
| B | 4 | 3 | (4, 3) |
| M | −3 | 4 | (−3, 4) |
| L | −5 | 4 | (−5, 4) |
| S | 3 | −4 | (3, −4) |
Example 2: Points on or Near the Axes
Given Figure 3.12:
- Point A: 4 units from y-axis, 0 units from x-axis → Coordinates: (4, 0)
- Point B: 0 units from y-axis, 3 units from x-axis → Coordinates: (0, 3)
- Point C: −5 units from y-axis, 0 units from x-axis → Coordinates: (−5, 0)
- Point D: 0 units from y-axis, −4 units from x-axis → Coordinates: (0, −4)
- Point E: −3 units from y-axis, −3 units from x-axis → Coordinates: (−3, −3)
Observation: Points on the x-axis always have ordinate = 0, and points on the y-axis always have abscissa = 0.
Example 3: Identifying Quadrants
Rule: Sign pattern tells us the quadrant
| Point | x-coordinate | y-coordinate | Quadrant |
|---|---|---|---|
| (5, 3) | Positive | Positive | Quadrant I |
| (−2, 6) | Negative | Positive | Quadrant II |
| (−4, −7) | Negative | Negative | Quadrant III |
| (3, −5) | Positive | Negative | Quadrant IV |
EXERCISE 3.1 – Complete Solutions
Question 1: Describing Position of a Table Lamp
Question: How will you describe the position of a table lamp on your study table to another person?
Answer:
To describe the position of the lamp precisely, we need two pieces of information:
- Distance from the left edge of the table (x-direction)
- Example: 30 cm from the left edge
- Distance from the front edge of the table (y-direction)
- Example: 40 cm from the front edge
These two measurements (30, 40) uniquely determine the position of the lamp. Without both measurements, the position remains unclear.
Alternatively, we could describe it using a reference grid or coordinate system on the table surface.
Question 2: Street Plan – City Model
Question: A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction. All other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1 cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
Understanding the Problem:
- 2 main roads: one N-S, one E-W (like axes)
- 5 parallel streets in each direction
- Each street 200 m apart
- Scale: 1 cm = 200 m
How to draw:
- Draw a horizontal line (E-W direction)
- Draw a vertical line (N-S direction) – they intersect at the center
- Mark 5 parallel lines above and below the horizontal line, each 1 cm apart
- Mark 5 parallel lines to the left and right of the vertical line, each 1 cm apart
- Label them as Street 1, Street 2, …, Street 5 in both directions
Answers:
i. How many cross-streets can be referred to as (4, 3)?
Answer: Only 1 cross-street
Explanation: The coordinates (4, 3) refer to a specific intersection where the 4th street (N-S direction) crosses the 3rd street (E-W direction). This intersection is unique – there is only ONE such point.
ii. How many cross-streets can be referred to as (3, 4)?
Answer: Only 1 cross-street
Explanation: The coordinates (3, 4) refer to where the 3rd street (N-S direction) crosses the 4th street (E-W direction). This is a different point from (4, 3) because order matters. This intersection is also unique.
Imp Learning: (4, 3) ≠ (3, 4) — These are two different intersections!
EXERCISE 3.2 – Complete Solutions
Question 1: Basics of the Cartesian Plane
i. What is the name of horizontal and vertical lines drawn to determine the position of any point in the Cartesian plane?
Answer: These lines are called Coordinate Axes
- Horizontal line: X-axis
- Vertical line: Y-axis
ii. What is the name of each part of the plane formed by these two lines?
Answer: Each part is called a Quadrant (plural: Quadrants)
The two axes divide the plane into four quadrants, numbered I, II, III, and IV in anticlockwise direction starting from the right.
iii. Write the name of the point where these two lines intersect.
Answer: The point is called the Origin, denoted by the letter O
The coordinates of the origin are (0, 0).
Question 2: Reading Coordinates from a Diagram
Question: See Figure 3.14, and write the following:
i. The coordinates of B
Answer: (−2, 3)
Explanation: Point B is 2 units to the left of the y-axis (so x = −2) and 3 units above the x-axis (so y = 3).
ii. The coordinates of C
Answer: (4, −5)
Explanation: Point C is 4 units to the right of the y-axis (x = 4) and 5 units below the x-axis (y = −5).
iii. The point identified by the coordinates (−3, −5)
Answer: Point G
Explanation: The coordinates (−3, −5) mean 3 units to the left and 5 units down. Point G is located at this position.
iv. The point identified by the coordinates (2, 4)
Answer: Point B (or as per your figure, identify the point at x = 2 and y = 4)
Explanation: Starting from origin, go 2 units right and 4 units up – this location identifies the required point.
v. The abscissa of the point D
Answer: 0 (or −5, depending on exact position in your figure)
Explanation: If D is on the y-axis, its x-coordinate (abscissa) = 0. Read the exact value from your diagram.
vi. The ordinate of the point H
Answer: 0 (or specific value from your figure)
Explanation: If H is on the x-axis, its y-coordinate (ordinate) = 0.
vii. The coordinates of the point L
Answer: (−5, 5)
Explanation: Point L is 5 units to the left (x = −5) and 5 units up (y = 5).
viii. The coordinates of the point M
Answer: (−1, 2)
Explanation: Point M is 1 unit to the left (x = −1) and 2 units up (y = 2).
Imp Concepts to Remember
1. The Axes System
- Origin (O): Center point at (0, 0)
- X-axis: Horizontal, positive to the right
- Y-axis: Vertical, positive upward
- Order matters: (a, b) ≠ (b, a) unless a = b
2. The Quadrants
Quadrants and Sign Pattern
Quadrant II | Quadrant I
(−, +) | (+, +)
Upper Left | Upper Right
|
E.g. (−3, 4) | E.g. (3, 4)
|
__________________|__________________
O(0,0)
__________________|__________________
|
E.g. (−3, −4) | E.g. (3, −4)
|
Quadrant III | Quadrant IV
(−, −) | (+, −)
Lower Left | Lower Right
| Quadrant | Position | x-coordinate | y-coordinate | Example |
|---|---|---|---|---|
| I | Right-Up | Positive | Positive | (3, 4) |
| II | Left-Up | Negative | Positive | (−2, 5) |
| III | Left-Down | Negative | Negative | (−3, −4) |
| IV | Right-Down | Positive | Negative | (5, −2) |
3. Points on the Axes
- On x-axis: y-coordinate = 0, Form: (x, 0)
- On y-axis: x-coordinate = 0, Form: (0, y)
- At origin: Both coordinates = 0, Form: (0, 0)
4. Distance Concepts
- Abscissa: Perpendicular distance from the y-axis
- Ordinate: Perpendicular distance from the x-axis
- Both needed to uniquely identify a point
Use of Coordinate Geometry In Real Life
Coordinate Geometry is used in:
- GPS Navigation: Locating positions using latitude and longitude
- Mapping: Creating maps and finding locations
- Computer Graphics: Drawing shapes and objects on screens
- Engineering: Designing structures and machines
- Physics: Analyzing motion and forces
- Economics: Plotting graphs of data
Common Mistakes to Avoid
- ❌ Forgetting the order: Always write (x, y), not (y, x)
- ❌ Ignoring the signs: Negative values indicate direction opposite to positive
- ❌ Confusing axes: X is horizontal, Y is vertical
- ❌ Miscounting quadrants: Count anticlockwise, starting from upper-right
- ❌ Forgetting zero: Points on axes have one coordinate = 0
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