Chapter 3: Coordinate Geometry

1. The Cartesian System

Coordinate Geometry helps us locate a point in a plane using two perpendicular reference lines. This system was developed by the French mathematician René Descartes.

Key Definitions

Origin (O): The point where the X-axis and Y-axis intersect. Coordinates: \( (0, 0) \).
X-axis: The horizontal number line (\( X'OX \)).
Y-axis: The vertical number line (\( Y'OY \)).

2. Quadrants & Sign Convention

The axes divide the plane into four parts called Quadrants. The sign of coordinates depends on the quadrant.

Quadrant	X-coordinate	Y-coordinate	Sign
I (First)	Positive (+)	Positive (+)	\((+, +)\)
II (Second)	Negative (-)	Positive (+)	\((-, +)\)
III (Third)	Negative (-)	Negative (-)	\((-, -)\)
IV (Fourth)	Positive (+)	Negative (-)	\((+, -)\)

2.1 Quadrant Logic (No Memorizing)

Just ask two questions:

Is x positive or negative?
Is y positive or negative?

(+, +) → Quadrant I
(−, +) → Quadrant II
(−, −) → Quadrant III
(+, −) → Quadrant IV

🧠 Trick

Move anticlockwise starting from (+,+).

Quadrant Hunter

Enter coordinates to find their home!

Waiting for input...

3. Understanding Coordinates

The position of a point is given by an ordered pair \( (x, y) \).

Term	Alternative Name	Definition
x-coordinate	Abscissa	Perpendicular distance from the Y-axis.
y-coordinate	Ordinate	Perpendicular distance from the X-axis.

Watch Out:
The coordinates \( (x, y) \) are NOT the same as \( (y, x) \) unless \( x = y \). That's why they are called "ordered pairs".

3.1 Why Are Coordinates Ordered?

Imagine this: You tell a friend your seat location 🎟️

“Row 3, Seat 5” is NOT the same as “Row 5, Seat 3”.

Similarly:

\( (3, 5) \) → Go 3 units right, then 5 units up
\( (5, 3) \) → Go 5 units right, then 3 units up

⚠️ CBSE Trap: Swapping coordinates gives a different point!

🧠 Think

Can two different ordered pairs represent the same point?

No. Order matters.

3.2 How to Plot a Point (CBSE Method)

To plot a point \( (x, y) \):

Start from the origin (0,0)
Move x units along X-axis (right if +, left if −)
From there, move y units parallel to Y-axis
Mark the point and label it

📘 Exam Tip

CBSE awards marks for correct steps even if the point is slightly misplaced.

4. Points on the Axes

What if a point lies exactly on a line?

Point on X-axis

If a point is on the X-axis, its distance from the X-axis is zero.

Therefore, its y-coordinate (ordinate) is 0.

Form: \( (x, 0) \)

Point on Y-axis

If a point is on the Y-axis, its distance from the Y-axis is zero.

Therefore, its x-coordinate (abscissa) is 0.

Form: \( (0, y) \)

4.1 Distance from Axes – Think Visually

Key Idea:

Distance from Y-axis → absolute value of x
Distance from X-axis → absolute value of y

Example: Point \( P(-4, 3) \)

Distance from Y-axis = 4 units
Distance from X-axis = 3 units

⚠️ Sign does NOT affect distance, only magnitude matters.

🌍 Coordinates in Real Life

📍 Google Maps (Latitude, Longitude)
🎮 Game maps & player position
📊 Graphs in science & economics
🏫 Classroom seating plans

Coordinate geometry helps computers, GPS, and games understand where things are.

Concept Mastery Quiz

1. The point \( (-3, 5) \) lies in which quadrant?

A) I Quadrant

B) II Quadrant

C) III Quadrant

2. The abscissa of the point \( (4, -7) \) is:

A) 4

B) -7

C) -3

3. A point on the Y-axis has the form:

A) \( (x, 0) \)

B) \( (0, y) \)

C) \( (y, y) \)

4. The perpendicular distance of point \( P(3, 4) \) from the Y-axis is:

A) 3 units

B) 4 units

C) 5 units

5. In which quadrant do both coordinates have a negative sign?

A) II Quadrant

B) III Quadrant

C) IV Quadrant

🚨 Common CBSE Mistakes

Writing (y, x) instead of (x, y)
Forgetting signs of coordinates
Confusing X-axis and Y-axis
Assuming points on axes belong to quadrants

Avoid these → Easy marks saved 🎯