📐 Coordinate Geometry - Class 9

Complete notes with diagrams, examples, and formula sheet

1. Introduction to Coordinate Geometry

Coordinate Geometry (also called Cartesian Geometry or Analytic Geometry) is a branch of mathematics that uses numbers to represent geometric figures and positions. It bridges algebra and geometry, allowing us to describe shapes using equations.

📖 What is Coordinate Geometry?

Coordinate Geometry is the study of geometry using a coordinate system. We use two perpendicular lines (axes) to locate any point on a plane using two numbers called coordinates.

This system was developed by the French mathematician René Descartes, which is why it's called the Cartesian system.

1.1 Why Study Coordinate Geometry?

It helps us represent geometric shapes using algebraic equations.
We can calculate distances, midpoints, and areas using simple formulas.
It's used in GPS systems, computer graphics, architecture, and engineering.
It provides a powerful tool to solve geometric problems algebraically.
Makes complex geometric proofs simpler using coordinate methods.

2. The Cartesian Plane

The Cartesian plane is formed by two perpendicular number lines that intersect at a point called the origin.

2.1 Components of Cartesian Plane

X-axis (Horizontal axis): The horizontal number line running left to right.
Y-axis (Vertical axis): The vertical number line running bottom to top.
Origin (O): The point where both axes intersect, represented as (0, 0).
Quadrants: The four regions created by the intersection of axes.
The axes divide the plane into four equal parts called quadrants.

📊 Cartesian Plane Structure

Imagine a "+" sign: the horizontal line is the X-axis, and the vertical line is the Y-axis.

The meeting point in the center is the Origin (0, 0).

Right = positive X, Left = negative X, Up = positive Y, Down = negative Y

2.2 The Four Quadrants

Quadrant	Position	X-coordinate	Y-coordinate	Example
I (First)	Top-Right	Positive (+)	Positive (+)	(3, 4)
II (Second)	Top-Left	Negative (-)	Positive (+)	(-2, 5)
III (Third)	Bottom-Left	Negative (-)	Negative (-)	(-4, -3)
IV (Fourth)	Bottom-Right	Positive (+)	Negative (-)	(5, -2)

🔑 Remember: Quadrant Signs

I: (+, +) → Both positive
II: (-, +) → X negative, Y positive
III: (-, -) → Both negative
IV: (+, -) → X positive, Y negative
Mnemonic: Start from top-right, go anticlockwise

3. Coordinates of a Point

Every point on the Cartesian plane can be uniquely identified by an ordered pair of numbers called coordinates.

📖 What are Coordinates?

Coordinates are a pair of numbers written as (x, y) that represent the position of a point on the plane.

x-coordinate (Abscissa): The distance from the Y-axis (horizontal distance).

y-coordinate (Ordinate): The distance from the X-axis (vertical distance).

Format: Point P = (x, y) where x comes first, then y.

3.1 Understanding Coordinates

The first number (x) tells how far to move horizontally from the origin.
The second number (y) tells how far to move vertically from the origin.
Positive x means move right; negative x means move left.
Positive y means move up; negative y means move down.
Order matters! (3, 5) and (5, 3) are different points.

📝 Example: Plotting Points

Q1: Plot the point A(4, 3)

Solution:

• Start at origin (0, 0)

• Move 4 units to the RIGHT (positive x)

• Then move 3 units UP (positive y)

• Mark the point A(4, 3) in Quadrant I

Q2: Plot the point B(-3, 2)

Solution:

• Start at origin (0, 0)

• Move 3 units to the LEFT (negative x)

• Then move 2 units UP (positive y)

• Mark the point B(-3, 2) in Quadrant II

3.2 Special Positions

Origin: (0, 0) - The center point where axes meet.
On X-axis: Points like (5, 0), (-3, 0) - Y-coordinate is 0.
On Y-axis: Points like (0, 4), (0, -2) - X-coordinate is 0.
Points on axes don't belong to any quadrant.
Distance from X-axis = |y-coordinate|
Distance from Y-axis = |x-coordinate|

📝 Example: Identifying Positions

Q: In which quadrant or on which axis do these points lie?

a) (5, 3) b) (-2, -4) c) (0, 5) d) (3, 0) e) (-3, 2)

Solution:

a) (5, 3) → Both positive → Quadrant I

b) (-2, -4) → Both negative → Quadrant III

c) (0, 5) → X is 0 → On Y-axis (above origin)

d) (3, 0) → Y is 0 → On X-axis (right of origin)

e) (-3, 2) → X negative, Y positive → Quadrant II

4. Plotting Points on Graph

Plotting means marking the exact position of a point on the coordinate plane based on its coordinates.

4.1 Steps to Plot a Point

Step 1: Draw two perpendicular axes (X and Y) intersecting at origin.
Step 2: Mark equal intervals on both axes (scale).
Step 3: Start from the origin (0, 0).
Step 4: Move horizontally according to x-coordinate (right if +, left if -).
Step 5: From there, move vertically according to y-coordinate (up if +, down if -).
Step 6: Mark the point and label it.

⚠️ Important: Choosing Scale

Always choose a suitable scale depending on the values of coordinates.

If coordinates are large (like 50, 100), use larger intervals (1 cm = 10 units).

If coordinates are small (like 2, 3), use smaller intervals (1 cm = 1 unit).

Keep the same scale on both axes unless specified otherwise.

5. Distance Between Two Points

In coordinate geometry, we can calculate the distance between any two points using a formula derived from the Pythagorean theorem.

📖 Distance Formula

The distance between two points P(x₁, y₁) and Q(x₂, y₂) is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This formula gives the straight-line distance between the two points.

5.1 Understanding the Formula

The formula is based on the Pythagorean theorem (a² + b² = c²).
(x₂ - x₁) represents the horizontal distance between points.
(y₂ - y₁) represents the vertical distance between points.
We square these distances, add them, and take the square root.
The result is always positive (distance cannot be negative).
Order doesn't matter: distance from A to B = distance from B to A.

📝 Example: Calculating Distance

Q1: Find the distance between A(2, 3) and B(5, 7)

Solution:

Here, x₁ = 2, y₁ = 3, x₂ = 5, y₂ = 7

Using distance formula:

d = √[(5-2)² + (7-3)²]

d = √[3² + 4²]

d = √[9 + 16]

d = √25

d = 5 units

Q2: Find distance between P(1, 2) and Q(4, 6)

Solution:

d = √[(4-1)² + (6-2)²]

d = √[3² + 4²]

d = √[9 + 16]

d = √25 = 5 units

Q3: Find distance from origin to point (3, 4)

Solution:

Origin is (0, 0), Point is (3, 4)

d = √[(3-0)² + (4-0)²]

d = √[9 + 16] = √25 = 5 units

5.2 Special Cases

Horizontal line: Points like (2, 3) and (5, 3) - Same y-coordinate, distance = |x₂ - x₁|
Vertical line: Points like (4, 2) and (4, 7) - Same x-coordinate, distance = |y₂ - y₁|
Distance from origin: For point (x, y), distance = √(x² + y²)
If distance is 0, both points are the same.

📝 Example: Special Cases

Q1: Distance between (2, 5) and (7, 5)

Solution: Same y-coordinate (horizontal line)

Distance = |7 - 2| = 5 units

Q2: Distance between (3, 1) and (3, 6)

Solution: Same x-coordinate (vertical line)

Distance = |6 - 1| = 5 units

6. Applications and Problem Solving

6.1 Verifying Geometric Shapes

We can verify if points form specific shapes using distance formula.
Equilateral triangle: All three sides equal.
Isosceles triangle: Two sides equal.
Right triangle: Use Pythagorean theorem: a² + b² = c²
Square: All sides equal and diagonals equal.
Rectangle: Opposite sides equal and diagonals equal.

📝 Example: Verifying Triangle Type

Q: Show that points A(0, 0), B(3, 0), and C(0, 4) form a right triangle.

Solution:

Calculate all three sides:

AB = √[(3-0)² + (0-0)²] = √9 = 3

BC = √[(0-3)² + (4-0)²] = √(9+16) = √25 = 5

CA = √[(0-0)² + (0-4)²] = √16 = 4

Check Pythagorean theorem: AB² + CA² = BC²?

3² + 4² = 9 + 16 = 25

5² = 25

Yes! It's a right triangle (right angle at A)

6.2 Collinearity of Points

Three points are collinear if they lie on the same straight line.
Check: If A, B, C are collinear, then AB + BC = AC (or any other combination).
If the sum of two distances equals the third, points are collinear.

📝 Example: Checking Collinearity

Q: Are points A(1, 2), B(3, 4), and C(5, 6) collinear?

Solution:

AB = √[(3-1)² + (4-2)²] = √(4+4) = √8 = 2√2

BC = √[(5-3)² + (6-4)²] = √(4+4) = √8 = 2√2

AC = √[(5-1)² + (6-2)²] = √(16+16) = √32 = 4√2

Check: AB + BC = 2√2 + 2√2 = 4√2 = AC ✓

Yes! The points are collinear.

7. Lines Parallel to Axes

Some lines on the coordinate plane are parallel to either the X-axis or Y-axis and have special properties.

7.1 Line Parallel to X-axis

A line parallel to X-axis has the same y-coordinate for all points.
Equation form: y = k (where k is a constant)
Example: y = 3 (all points have y-coordinate = 3)
Points on this line: (1, 3), (2, 3), (-5, 3), etc.
The X-axis itself is y = 0.

7.2 Line Parallel to Y-axis

A line parallel to Y-axis has the same x-coordinate for all points.
Equation form: x = k (where k is a constant)
Example: x = 2 (all points have x-coordinate = 2)
Points on this line: (2, 1), (2, 5), (2, -3), etc.
The Y-axis itself is x = 0.

⚠️ Remember

Parallel to X-axis: y-coordinate is constant → Horizontal line

Parallel to Y-axis: x-coordinate is constant → Vertical line

Perpendicular lines: One parallel to X-axis, other to Y-axis

8. Real-Life Applications

8.1 Where is Coordinate Geometry Used?

GPS and Navigation: Location tracking uses coordinates (latitude, longitude).
Computer Graphics: Every pixel on screen has coordinates.
Architecture: Building designs use coordinate systems for precision.
Maps: Cities and locations are marked using coordinates.
Video Games: Character positions tracked using coordinates.
Robotics: Robot movements programmed using coordinate geometry.
Aviation: Flight paths calculated using coordinate systems.

📚 Quick Formula Sheet - Coordinate Geometry

Basic Definitions

Point: P(x, y)

x = Abscissa (horizontal)

y = Ordinate (vertical)

Origin: O(0, 0)

Order matters: (x, y) ≠ (y, x)

Quadrants

I: (+, +) - Top Right

II: (-, +) - Top Left

III: (-, -) - Bottom Left

IV: (+, -) - Bottom Right

Axes points: Not in any quadrant

Distance Formula

Between P(x₁,y₁) and Q(x₂,y₂):

d = √[(x₂-x₁)² + (y₂-y₁)²]

From Origin:

d = √(x² + y²)

Based on Pythagorean theorem

Special Distances

Horizontal line:

Points (x₁,y) & (x₂,y): d = |x₂-x₁|

Vertical line:

Points (x,y₁) & (x,y₂): d = |y₂-y₁|

Same coordinate simplifies formula

Position Properties

Distance from X-axis = |y|

Distance from Y-axis = |x|

On X-axis: y = 0

On Y-axis: x = 0

Absolute value gives distance

Lines Parallel to Axes

Parallel to X-axis:

y = k (constant)

Parallel to Y-axis:

x = k (constant)

k can be any real number

Triangle Verification

Right Triangle:

a² + b² = c² (Pythagoras)

Equilateral:

All sides equal: AB = BC = CA

Isosceles: Two sides equal

Collinearity Test

Points A, B, C collinear if:

AB + BC = AC (or similar)

Sum of two distances = third distance

All three points on same line

💡 Study Tips for Coordinate Geometry

• Always draw a rough diagram before solving problems - visualization helps!

• Remember the quadrant signs: I(+,+), II(-,+), III(-,-), IV(+,-)

• In distance formula, squaring removes the sign, so order doesn't matter.

• Use graph paper for accurate plotting and measurement.

• Check your answers: distance should always be positive.

• Practice plotting points daily to become comfortable with the coordinate system.

• For large numbers, choose an appropriate scale to fit the graph.

• Remember: Horizontal = X-axis, Vertical = Y-axis

🔑 Common Mistakes to Avoid

Don't confuse (x, y) with (y, x) - order is crucial!
Don't forget to take square root in distance formula.
Remember to square the differences: (x₂-x₁)², not x₂²-x₁²
Check which quadrant - sign errors are common.
On axes, one coordinate is always zero.
Distance cannot be negative - use absolute value if needed.