Coordinate Geometry Class 10 Full Chapter with Formulas and Examples

Coordinate Geometry is one of the most important chapters in Class 10 Maths because it connects geometry with algebra in a very simple and practical way. In this chapter, students learn how to represent points on a plane, how to calculate the distance between two points, how to divide a line segment in a given ratio, and how to solve board-level questions using formulas directly.

This chapter is also considered one of the most scoring areas in the exam because the concepts are direct and the formulas are limited. Once students understand the Cartesian plane, coordinate notation, distance formula, and section formula properly, most questions become easy and manageable.

At Deeksha Vedantu, we always encourage students to study Coordinate Geometry with a visual approach first and then move to formulas. When students understand where the point lies and what the question is asking geometrically, the calculations become much easier.

Why Coordinate Geometry Is Important in Class 10

Coordinate Geometry is important because it combines visual understanding with formula-based solving.

Why Students Should Prepare This Chapter Well

• it is a regular board-exam chapter
• it is highly scoring when formulas are clear
• it helps students improve coordinate-based reasoning
• it includes direct, objective, and application-based questions
• it builds a strong base for higher Maths

Chapter Overview at a Glance

This quick table helps students revise the full chapter faster.

Quick Concept Table

What Is Coordinate Geometry

Coordinate Geometry is the branch of mathematics in which the position of points is studied using numbers on a plane.

Instead of only drawing shapes, we describe their position using coordinates.

Cartesian Coordinate System

The full chapter begins with the Cartesian coordinate system.

Meaning of the Cartesian Plane

A Cartesian coordinate system is formed by two perpendicular number lines:

• x-axis
• y-axis

These two lines intersect at one point and help us locate any point on the plane.

X-Axis and Y-Axis Table

Origin

The point where the x-axis and y-axis intersect is called the origin.

Coordinates of Origin

The coordinates of the origin are:

(0, 0)

This is one of the most basic and important facts of the chapter.

Coordinates of a Point

A point on the Cartesian plane is represented by an ordered pair.

General Form

A point is written as:

(x, y)

Meaning of the Ordered Pair

• the first value shows the x-coordinate
• the second value shows the y-coordinate

This order must never be changed.

Important Rule About Coordinates

Whenever you write a coordinate:

• x-coordinate always comes first
• y-coordinate always comes second

For example:

(2, -3)

means:

• x = 2
• y = -3

It does not mean x = -3 and y = 2.

Quadrants in the Cartesian Plane

The x-axis and y-axis divide the plane into four parts. These four parts are called quadrants.

Quadrant Summary Table

Important Note About Quadrants

Points lying on the axes are not part of any quadrant.

Points on the Axes

This is a very important concept for board questions.

Point on the X-Axis

If a point lies on the x-axis, then its y-coordinate is always 0.

General Form

(x, 0)

Point on the Y-Axis

If a point lies on the y-axis, then its x-coordinate is always 0.

General Form

(0, y)

Axis-Based Point Summary Table

Distance Formula

This is the first major formula in the chapter.

The distance formula is used to find the distance between two points in the Cartesian plane.

Formula for Distance Between Two Points

If the coordinates of two points are:

(x₁, y₁) and (x₂, y₂)

then the distance between them is:

√[(x₂ – x₁)² + (y₂ – y₁)²]

This is one of the most important formulas in the chapter.

Why the Distance Formula Is Important

This formula helps students:

• find the length of a line segment
• check whether two distances are equal
• prove whether a triangle is isosceles or right-angled
• solve assertion-reason and competency-based questions

Easy Rule While Using Distance Formula

A simple way to avoid confusion is:

• whichever point you treat as the second point, use both its coordinates as x₂ and y₂
• whichever point you treat as the first point, use both its coordinates as x₁ and y₁

Even if you reverse the order of the points, the answer remains the same.

Solved Examples Based on Distance Formula

Solved Example 1: Distance Between Two Simple Points

Find the distance between A(2, 3) and B(-2, -4).

Step 1: Use the Formula

Distance AB = √[(-2 – 2)² + (-4 – 3)²]

Step 2: Simplify

AB = √[(-4)² + (-7)²]

AB = √[16 + 49]

AB = √65

Answer

The distance between the points is √65 units.

Solved Example 2: Trigonometric Coordinate Question

Find the distance between A(tan θ, 0) and B(1, √(2tan θ)).

Step 1: Use the Distance Formula

AB = √[(1 – tan θ)² + (√(2tan θ) – 0)²]

Step 2: Expand and Simplify

AB = √[(1 – 2tan θ + tan² θ) + 2tan θ]

AB = √(1 + tan² θ)

Step 3: Use Identity

1 + tan² θ = sec² θ

So:

AB = √(sec² θ) = sec θ

Answer

The distance between the points is sec θ.

Equidistant Point Concept

A point is said to be equidistant from two points if its distance from both points is the same.

Mathematical Meaning

If P is equidistant from A and B, then:

PA = PB

This idea is used in many board questions.

Solved Example 3: Find k Using Equidistant Condition

Find the value of k if the point P(0, 2) is equidistant from A(3, k) and B(k, 5).

Step 1: Use the Equidistant Idea

Since P is equidistant from A and B:

PA = PB

Step 2: Write Both Distances

PA = √[(3 – 0)² + (k – 2)²]

PB = √[(k – 0)² + (5 – 2)²]

Step 3: Equate and Simplify

√[9 + (k – 2)²] = √[k² + 9]

Squaring both sides:

9 + (k – 2)² = k² + 9

9 + k² + 4 – 4k = k² + 9

13 – 4k = 9

4 = 4k

k = 1

Answer

The value of k is 1.

Distance of a Point from the Axes

This is a very common short-answer or objective question type.

Distance from the Y-Axis

The distance of a point from the y-axis is the absolute value of its x-coordinate.

Example

For the point (2, 3), the distance from the y-axis is 2 units.

Distance from the X-Axis

The distance of a point from the x-axis is the absolute value of its y-coordinate.

Example

For the point (2, 3), the distance from the x-axis is 3 units.

Axis Distance Summary Table

Solved Example 4: Distance from the Y-Axis

Find the shortest distance of the point (2, 3) from the y-axis.

Solution

The distance from the y-axis is the x-coordinate in absolute value.

So:

Distance = 2 units

Answer

The shortest distance from the y-axis is 2 units.

Section Formula

This is the second major formula of the chapter.

The section formula is used when a point divides the line segment joining two points in a given ratio.

Internal Division in Class 10

In Class 10, students mainly study internal division. This means the point lies between the two given points.

Formula for Section Formula

If a point P(x, y) divides the line joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂, then:

x = (m₁x₂ + m₂x₁)/(m₁ + m₂)

y = (m₁y₂ + m₂y₁)/(m₁ + m₂)

This is a very important formula and must be memorised carefully.

Easy Memory Trick for Section Formula

When using m₁, multiply it with the farther coordinate.

When using m₂, multiply it with the other coordinate.

That is why:

• m₁ goes with x₂ and y₂
• m₂ goes with x₁ and y₁

Why Section Formula Is Important

It helps students:

• find the coordinates of the dividing point
• find the ratio in which a line is divided
• solve axis-based intersection questions
• solve board and competency-style questions

Solved Examples Based on Section Formula

Solved Example 5: Find a Dividing Point

If the point P(k, 0) divides the line segment joining A(2, -2) and B(-7, -4) in the ratio 1 : 2, find the value of k.

Step 1: Use the Section Formula for x-coordinate

k = [1 × (-7) + 2 × 2]/(1 + 2)

Step 2: Simplify

k = (-7 + 4)/3

k = -3/3

k = -1

Answer

The value of k is -1.

Solved Example 6: Find the Ratio in Which X-Axis Divides a Line Segment

In what ratio does the x-axis divide the line segment joining P(3, -6) and Q(5, 3)?

Step 1: Use the Key Idea

Since the dividing point lies on the x-axis, its y-coordinate is 0.

Step 2: Let the Ratio Be m₁ : m₂

Using the section formula for y-coordinate:

0 = (m₁ × 3 + m₂ × -6)/(m₁ + m₂)

Step 3: Simplify

0 = 3m₁ – 6m₂

3m₁ = 6m₂

m₁ = 2m₂

So:

m₁ : m₂ = 2 : 1

Answer

The x-axis divides the line segment in the ratio 2 : 1.

Solved Example 7: Point on X-Axis Equidistant from Two Given Points

Find the point on the x-axis which is equidistant from A(2, -5) and B(-2, 9).

Step 1: Let the Required Point Be P(x, 0)

Since the point lies on the x-axis, its y-coordinate must be 0.

Step 2: Use Equidistant Condition

PA = PB

Step 3: Write the Distances

PA = √[(2 – x)² + (-5 – 0)²]

PB = √[(-2 – x)² + (9 – 0)²]

Step 4: Equate and Simplify

√[(2 – x)² + 25] = √[(-2 – x)² + 81]

Squaring both sides:

(2 – x)² + 25 = (-2 – x)² + 81

4 + x² – 4x + 25 = 4 + x² + 4x + 81

29 – 4x = 85 + 4x

-56 = 8x

x = -7

Answer

The required point is (-7, 0).

Triangle Verification Questions in Coordinate Geometry

These are very important board-style questions because they combine distance formula with geometric properties.

Solved Example 8: Prove That Three Points Form a Right-Angled Isosceles Triangle

Prove that the points A(3, 0), B(6, 4), and C(-1, 3) are the vertices of a right-angled isosceles triangle.

Step 1: Find AB

AB = √[(6 – 3)² + (4 – 0)²]

AB = √[9 + 16] = 5

Step 2: Find AC

AC = √[(-1 – 3)² + (3 – 0)²]

AC = √[16 + 9] = 5

So:

AB = AC

This proves the triangle is isosceles.

Step 3: Find BC

BC = √[(-1 – 6)² + (3 – 4)²]

BC = √[49 + 1] = √50

Step 4: Apply Pythagoras Check

AB² = 25

AC² = 25

BC² = 50

Since:

AB² + AC² = BC²

25 + 25 = 50

So the triangle is right-angled.

Answer

The triangle formed is a right-angled isosceles triangle.

Common Board Question Patterns from Coordinate Geometry

This chapter usually gives repeated types of questions.

Case 1: Direct Distance Formula Question

Students are asked to find the distance between two points.

Case 2: Equidistant Point Question

Students are asked to find an unknown value using equal distance condition.

Case 3: Distance from Axis Question

These ask for the shortest distance of a point from the x-axis or y-axis.

Case 4: Section Formula for Coordinates

These ask for the coordinates of a point dividing a segment.

Case 5: Section Formula for Ratio

These ask in what ratio a line is divided by an axis or a point.

Case 6: Triangle Verification Question

Students may be asked to prove whether points form an isosceles, right-angled, or right-angled isosceles triangle.

Board Pattern Summary Table

Common Mistakes Students Make in Coordinate Geometry

These mistakes are very common in board exams.

Common Mistakes Table

Quick Revision Formula Sheet

This section is useful for final exam revision.

Formula Sheet Table

Best Study Strategy for Coordinate Geometry

Coordinate Geometry becomes easy when students revise it in the right order.

Step-by-Step Revision Table

Practice Questions for Students

Important Practice Questions

• Find the distance between (2, 3) and (-2, -4).
• Find the shortest distance of (5, -2) from the x-axis and y-axis.
• Find the value of k if P(0, 2) is equidistant from A(3, k) and B(k, 5).
• Find the coordinates of a point dividing the line segment joining two given points in the ratio 2 : 3.
• In what ratio does the x-axis divide the line joining two points?
• Prove whether three points form a right-angled triangle.

FAQs

Q1. What is Coordinate Geometry in Class 10?

Coordinate Geometry is the branch of mathematics in which points are represented on a plane using ordered pairs such as (x, y).

Q2. What is the Cartesian coordinate system?

It is a system formed by two perpendicular number lines called the x-axis and y-axis.

Q3. What are the coordinates of the origin?

The coordinates of the origin are (0, 0).

Q4. What is the distance formula in Coordinate Geometry?

The distance formula is √[(x₂ – x₁)² + (y₂ – y₁)²].

Q5. What is the section formula in Class 10?

If a point divides a line internally in the ratio m₁ : m₂, then its coordinates are found using:

x = (m₁x₂ + m₂x₁)/(m₁ + m₂)

y = (m₁y₂ + m₂y₁)/(m₁ + m₂)

Q6. What is the coordinate of a point on the x-axis?

A point on the x-axis has the form (x, 0).

Q7. What is the coordinate of a point on the y-axis?

A point on the y-axis has the form (0, y).

Q8. How can I score well in Coordinate Geometry?

You can score well by memorising the formulas, being careful with signs, and practising distance, section, and axis-based questions regularly.

Conclusion

Coordinate Geometry is one of the most direct and scoring chapters in Class 10 Maths because it uses only a few formulas but applies them in many smart ways. Once students understand the Cartesian plane, coordinate notation, distance formula, and section formula, the chapter becomes very clear and comfortable.

The best way to prepare this chapter is to revise the formulas daily, solve diagram-based examples, and practise negative-coordinate questions carefully. At Deeksha Vedantu, we always remind students that Coordinate Geometry becomes easy when the visual idea is understood first and the formula is applied calmly after that.