CLASS 9 :CHAPTER -1(

📐 Mathematics

Orienting Yourself:
The Use of Coordinates

Chapter 3 — Coordinate Geometry

📚 Class 9 🎯 NCERT / CBSE ✍️ Topper Notes 🔖 EnglishWithMRK Style

① What is Coordinate Geometry?

📖 Definition: Coordinate Geometry is the branch of mathematics that uses a number system (coordinates) to describe the exact position of a point in a plane (flat surface).

• Also called Cartesian Geometry — named after French mathematician René Descartes.
• It connects Algebra and Geometry together.
• It helps us locate any point on a flat surface using just two numbers.

🌍 Real-Life: Finding a seat in a cinema (Row D, Seat 7), locating a city on a map using latitude & longitude, or playing Battleship game — all use coordinates!

② The Cartesian Plane

📖 Definition: The Cartesian Plane (also called the Coordinate Plane) is a flat surface formed by drawing two number lines perpendicular (at 90°) to each other.

• X-axis → Horizontal number line (goes left ← and right →)
• Y-axis → Vertical number line (goes up ↑ and down ↓)
• Origin (O) → Point where X-axis and Y-axis meet = (0, 0)
• The plane is divided into 4 parts called Quadrants.

📊 Diagram: The Cartesian Plane

Y-axis ↑ | Q II | Q I (–, +) 4 | (+, +) 3 | 2 | • P(3,2) 1 | ←————————————0————+————————————→ X-axis -4-3-2-1 1 2 3 4 -1 | Q III -2 | Q IV (–, –) -3 | (+, –) -4 | ↓

📊 Quadrant Sign Chart

Q II
(−x, +y)

Q I
(+x, +y)

Q III
(−x, −y)

Q IV
(+x, −y)

← X-axis → ↑ Y-axis ↓

③ The Four Quadrants

Quadrant	Position	x sign	y sign	Example
Q I (First)	Top-Right	+ (Positive)	+ (Positive)	(3, 4)
Q II (Second)	Top-Left	− (Negative)	+ (Positive)	(−2, 5)
Q III (Third)	Bottom-Left	− (Negative)	− (Negative)	(−3, −1)
Q IV (Fourth)	Bottom-Right	+ (Positive)	− (Negative)	(5, −2)

🧠 Memory Trick for Signs: "All Students Take Calculus" — going anti-clockwise from Q I:
All (+,+) → Sine only (+,−)? No — try: "In Q1: ALL positive, Q2: x negative, Q3: ALL negative, Q4: y negative"
Or just remember: Quadrant I → ALL PLUS. Move anti-clockwise → one sign changes each time.

④ Coordinates of a Point

📖 Definition: The position of any point in the plane is written as an Ordered Pair (x, y) where x = Abscissa (horizontal distance from origin) and y = Ordinate (vertical distance from origin).

⭐ KEY FORMULA — Ordered Pair

Point P = (x, y) = (Abscissa, Ordinate)

Always write x first, then y  |  Order MATTERS — (3,2) ≠ (2,3)

• Abscissa (x) → Horizontal distance from Y-axis (left/right)
• Ordinate (y) → Vertical distance from X-axis (up/down)
• If a point is on the X-axis → y = 0 → coordinates are (x, 0)
• If a point is on the Y-axis → x = 0 → coordinates are (0, y)
• The Origin O → coordinates are (0, 0)

💡 Easy Way to Remember: "x comes before y in the alphabet, and x comes first in the ordered pair too!" Also: x = corridor (left-right), y = lift (up-down).

⑤ How to Plot a Point — Step by Step

• Step 1: Draw the X-axis (horizontal) and Y-axis (vertical).
• Step 2: Mark the Origin (0,0) where they cross.
• Step 3: Mark numbers on both axes (equal spacing).
• Step 4: Take the x-value → move that many units left or right from origin.
• Step 5: From there, take the y-value → move that many units up or down.
• Step 6: Put a dot and label it.

✏️ Example: Plot point A(−3, 2)
→ Start at Origin (0,0)
→ Move 3 units to the LEFT (x = −3)
→ Move 2 units UP (y = +2)
→ Mark and label the point A(−3, 2) → it lies in Quadrant II ✅

⑥ Special Positions of Points

Condition	Meaning	Example
y = 0	Point is on the X-axis	(4, 0), (−2, 0)
x = 0	Point is on the Y-axis	(0, 3), (0, −5)
x = 0, y = 0	Point is at the Origin	(0, 0)
x = y	Point is on the line y = x	(2, 2), (−3, −3)
x = −y	Point is on the line y = −x	(3, −3), (−1, 1)

⑦ 🔴 Important Formulas (Highlighted)

📍 Distance from Origin

d = √(x² + y²)

Distance of point (x, y) from Origin (0, 0)

📍 Distance Between Two Points

d = √[(x₂−x₁)² + (y₂−y₁)²]

Distance between A(x₁, y₁) and B(x₂, y₂) — for Class 10 but good to know!

📍 Midpoint Formula

M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

Midpoint of segment joining A(x₁, y₁) and B(x₂, y₂)

📍 Area of Triangle (given 3 vertices)

Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Use this to check if 3 points are collinear → Area = 0 means collinear!

⑧ Important Differences

Abscissa vs Ordinate

Abscissa (x)	Ordinate (y)
Horizontal distance from Y-axis	Vertical distance from X-axis
First number in the pair	Second number in the pair
+ve → right of Y-axis; −ve → left	+ve → above X-axis; −ve → below
On X-axis: x = any value	On Y-axis: y = any value

X-axis vs Y-axis

X-axis	Y-axis
Horizontal line	Vertical line
y = 0 for all points on it	x = 0 for all points on it
Divides plane into upper & lower	Divides plane into left & right
Goes left (−) and right (+)	Goes down (−) and up (+)

⑨ Collinear Points

📖 Definition: Three or more points are Collinear if they all lie on the same straight line.

• To check collinearity → Calculate Area of Triangle formed by the 3 points.
• If Area = 0 → Points are Collinear ✅
• If Area ≠ 0 → Points are NOT Collinear ❌

✏️ Example: Are A(1,1), B(2,2), C(3,3) collinear?
Area = ½|1(2−3) + 2(3−1) + 3(1−2)|
= ½|−1 + 4 − 3| = ½|0| = 0
∴ Points ARE collinear ✅ (They all lie on line y = x)

⑩ Reading Coordinates from a Graph

• To find coordinates of a point from graph: Draw perpendiculars to both axes.
• The foot of perpendicular on X-axis → gives the x-coordinate (abscissa).
• The foot of perpendicular on Y-axis → gives the y-coordinate (ordinate).
• Always read x first, then y.

📊 How to Read Coordinates

Y 5 | 4 | • P 3 | | ↖ P = (3, 4) 2 | | Read x=3 from X-axis 1 | | Read y=4 from Y-axis +—————————+——→ X 0 1 2 3 4

---

⑪ ✅ Exam Tips & Tricks

💡 Trick 1: "Walk then Climb" — First move horizontally (x), then vertically (y) to plot any point.

💡 Trick 2: On X-axis → y is always 0. On Y-axis → x is always 0. Don't forget!

🎯 Trick 3: Origin = (0,0). It belongs to NO quadrant — it's where axes meet.

🎯 Trick 4: Axes also belong to NO quadrant. Points on axes are NOT in any quadrant.

💡 Trick 5: (2,3) ≠ (3,2) — Order MATTERS! This is why it's called an "ordered" pair.

💡 Trick 6: Collinear check → Area = 0. If exam asks "are these collinear?" → use area formula!

🧠 Quadrant Mnemonic: Going anti-clockwise from Q I → "All Students Take Calculus"
Q I = All positive (+,+)  |  Q II = Sine positive (–,+)  |  Q III = Tangent positive (–,–)  |  Q IV = Cosine positive (+,–)

⑫ 🌍 Real-Life Examples

• 🗺️ Maps & GPS: Latitude (y) and Longitude (x) are coordinates that locate any city on Earth.
• ♟️ Chess: Each square has a coordinate (e.g., E4) — column = x, row = y.
• 🎮 Video Games: Every character position is stored as (x, y) coordinates on the screen.
• 🏥 Hospital Bed System: Ward D, Bed 7 → like coordinates (D, 7).
• 📱 Touchscreen: When you touch your phone screen, it records (x, y) pixel coordinates.
• 📡 Radar: Aircraft positions are tracked using coordinate systems from a central point.

⑬ ⚠️ Important Points — Don't Forget!

• Axes divide the plane into 4 quadrants — numbered anti-clockwise: I, II, III, IV.
• Points on axes are NOT in any quadrant.
• The Origin (0,0) is NOT in any quadrant.
• The x-coordinate tells LEFT/RIGHT, the y-coordinate tells UP/DOWN.
• Positive x → Right of Y-axis  |  Negative x → Left of Y-axis
• Positive y → Above X-axis  |  Negative y → Below X-axis
• A point with both coordinates equal but opposite sign (like (3,−3)) lies on the line y = −x.
• Always use a scale when drawing graphs — equal spacing on both axes.

⚡ Quick Revision — Last Minute Notes

• ✅ Coordinate Geometry = Algebra + Geometry using number pairs
• ✅ X-axis = Horizontal, Y-axis = Vertical
• ✅ Origin (O) = (0, 0) — where axes meet
• ✅ Ordered pair = (x, y) = (Abscissa, Ordinate)
• ✅ Q I: (+,+) | Q II: (−,+) | Q III: (−,−) | Q IV: (+,−)
• ✅ Quadrants numbered anti-clockwise from top-right
• ✅ On X-axis: y = 0 | On Y-axis: x = 0 | At Origin: x = y = 0
• ✅ Axes & Origin do NOT belong to any quadrant
• ✅ (3,4) ≠ (4,3) — order matters in a coordinate pair
• ✅ Collinear points → Area of triangle = 0
• ✅ Distance from origin = √(x²+y²)
• ✅ Plot: Move right/left first (x), then up/down (y)

✦ Made with ❤️ for exam success  |  EnglishWithMRK Style Notes ✦

Page 1 of 1  |  Class 9 Maths – Coordinate Geometry