CH – 7 COORDINATE GEOMETRY

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Chapter Overview

In this chapter, you will learn how to connect algebra with geometry using coordinates. You will cover:

• Understanding coordinates: Locating any point on a flat grid using a pair of numbers.
• Points on a coordinate plane: Working with positions on the horizontal and vertical axes.
• Distance between two points: Finding the exact length between any two points using their coordinates.
• Section formula: Calculating the coordinates of a point that divides a line segment in a given ratio.
• Finding the midpoint: Locating the point that lies exactly in the middle of a line segment.
• Using coordinates in geometry problems: Checking properties of geometric figures like triangles, squares, and parallelograms using formulas.

🎯 Board Exam Focus

Practice the following core skills carefully for board exam preparation:

• Reading coordinates correctly: Writing points in the standard (x, y) format without switching the numbers.
• Identifying x-coordinate and y-coordinate: Knowing the horizontal distance (abscissa) and vertical distance (ordinate).
• Finding the distance between two points: Applying the distance formula accurately with positive and negative numbers.
• Finding the midpoint of two points: Calculating the average of x-coordinates and y-coordinates.
• Using the section formula: Finding unknown coordinates or finding the ratio k : 1 in division problems.
• Solving questions based on coordinates: Checking if points form collinear lines, right triangles, squares, or parallelograms.

Most Important Concepts

What is Coordinate Geometry?

Coordinate geometry is a branch of mathematics where we use numbers and algebraic equations to study geometry.

In simple words, it helps us find the exact position of any point on a flat surface using a pair of numbers.

Think of it like locating a seat in a cinema hall. You need to know the row number and the seat number. Similarly, coordinate geometry uses two fixed perpendicular lines on a flat surface (called the coordinate plane) to mark exact locations.

Key parts of the coordinate plane:

• x-axis: The horizontal number line that runs left to right.
• y-axis: The vertical number line that runs up and down.
• Origin: The point where the x-axis and y-axis cross each other. It is represented by O and its coordinates are always (0, 0).
• Coordinates of a point: The pair of numbers (x, y) that tells us the exact location of a point.

Understanding Coordinates

Any point on a coordinate plane is written inside brackets as (x, y).

• The first number (x): Called the x-coordinate or abscissa. It tells you how far to move horizontally (left or right) from the y-axis.
• The second number (y): Called the y-coordinate or ordinate. It tells you how far to move vertically (up or down) from the x-axis.

Simple Examples:

• Point A(3, 5): Move 3 units to the right along the x-axis, and then 5 units straight up.
• Point B(−4, 2): Move 4 units to the left along the x-axis, and then 2 units straight up.
• Point C(6, 0): Lies directly on the x-axis at a distance of 6 units to the right of the origin.
• Point D(0, −3): Lies directly on the y-axis at a distance of 3 units down from the origin.
• Origin (0, 0): The starting point where you move 0 units right/left and 0 units up/down.

Distance Between Two Points

Finding the distance between two points means calculating the straight-line length connecting them.

When is this formula useful?

• To calculate the distance between two towns or locations on a map.
• To check if three points lie on the same straight line (collinear points).
• To identify geometric shapes like equilateral triangles, isosceles triangles, squares, or rhombuses using side lengths.

Step-by-Step Example:

Let us find the distance between point P(2, 3) and point Q(6, 6).

1. Identify the coordinates:
   • x₁ = 2, y₁ = 3
   • x₂ = 6, y₂ = 6
2. Use the Distance Formula:
   • Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
3. Substitute the values into the formula:
   • Distance PQ = √[(6 − 2)² + (6 − 3)²]
4. Subtract inside the brackets:
   • 6 − 2 = 4
   • 6 − 3 = 3
   • Distance PQ = √[(4)² + (3)²]
5. Square the numbers:
   • 4² = 16
   • 3² = 9
   • Distance PQ = √(16 + 9)
6. Add the values and take the square root:
   • Distance PQ = √25
   • Distance PQ = 5 units

The straight-line distance between P and Q is 5 units.

🧠 Easy Steps for Using the Distance Formula

1. Write both coordinates clearly: Mark the first point as (x₁, y₁) and the second point as (x₂, y₁).
2. Identify x-values and y-values: Keep them organized so you do not mix x with y.
3. Put values into the formula: Use the formula √[(x₂ − x₁)² + (y₂ − y₁)²].
4. Simplify inside the brackets first: Complete the subtractions before squaring.
5. Square both numbers: Remember that squaring any negative number always gives a positive result.
6. Add the two squared numbers: Combine the results inside the square root sign.
7. Find the final square root: Simplify the square root to get your final distance in units.

⚠️ Common Mistake Warning: Be extra careful when subtracting negative numbers! For example, if x₂ = 4 and x₁ = −3, then x₂ − x₁ becomes 4 − (−3) = 4 + 3 = 7. Never drop the double negative sign!

Section Formula

The section formula is used to find the coordinates of a point that cuts or divides a line segment joining two given points into a specific ratio.

Key Ideas:

• Internal Division: The dividing point lies inside on the line segment between the two endpoints.
• Ratio (m₁ : m₂): Shows how the line is shared into two parts. For example, a ratio of 1 : 2 means one part is short and the second part is twice as long.

Step-by-Step Example:

Find the coordinates of point P that divides the line segment joining A(1, 3) and B(7, 9) internally in the ratio 1 : 2.

1. Identify the given values:
   • x₁ = 1, y₁ = 3
   • x₂ = 7, y₂ = 9
   • m₁ = 1, m₂ = 2
2. Write the Section Formula:
   • x = (m₁x₂ + m₂x₁) ÷ (m₁ + m₂)
   • y = (m₁y₂ + m₂y₁) ÷ (m₁ + m₂)
3. Calculate the x-coordinate:
   • x = [(1 × 7) + (2 × 1)] ÷ (1 + 2)
   • x = (7 + 2) ÷ 3
   • x = 9 ÷ 3
   • x = 3
4. Calculate the y-coordinate:
   • y = [(1 × 9) + (2 × 3)] ÷ (1 + 2)
   • y = (9 + 6) ÷ 3
   • y = 15 ÷ 3
   • y = 5
5. Final Answer: The coordinates of point P are (3, 5).

Midpoint Formula

The midpoint of a line segment is the point that lies exactly in the middle, dividing the line segment into two equal halves (ratio 1 : 1).

To find the midpoint, you simply take the average of the two x-coordinates and the average of the two y-coordinates.

Step-by-Step Example:

Find the midpoint M of the line segment joining A(4, 8) and B(8, 2).

1. Identify the given coordinates:
   • x₁ = 4, y₁ = 8
   • x₂ = 8, y₂ = 2
2. Use the Midpoint Formula:
   • Midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
3. Calculate the x-coordinate:
   • x = (4 + 8) ÷ 2
   • x = 12 ÷ 2
   • x = 6
4. Calculate the y-coordinate:
   • y = (8 + 2) ÷ 2
   • y = 10 ÷ 2
   • y = 5
5. Final Answer: The midpoint M is (6, 5).

🧠 How to Know Which Formula to Use?

• Use the Distance Formula when the question asks for length, perimeter, distance, or asks you to show if points form a triangle, square, rhombus, or collinear line.
• Use the Midpoint Formula when you need to find the center of a circle, middle point of a side, or the point where diagonals of a parallelogram bisect each other.
• Use the Section Formula when a question mentions a point dividing a line in a given ratio (like 2 : 3 or 1 : 2), or asks you to find points of trisection (dividing into 3 equal parts).

📌 Important Formulas & Results

• Coordinates of the Origin:(0, 0)
  
• Point on the x-axis:(x, 0)
  
• Point on the y-axis:(0, y)
  
• Distance between two points P(x₁, y₁) and Q(x₂, y₂):PQ = √[(x₂ − x₁)² + (y₂ − y₁)²]
• Distance of a point P(x, y) from the origin (0, 0):OP = √(x² + y²)
• Section Formula:P(x, y) = ((m₁x₂ + m₂x₁) ÷ (m₁ + m₂), (m₁y₂ + m₂y₁) ÷ (m₁ + m₂))
• Section Formula with ratio k : 1:P(x, y) = ((kx₂ + x₁) ÷ (k + 1), (ky₂ + y₁) ÷ (k + 1))
• Midpoint Formula:M(x, y) = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

🧠 Concepts Students Find Difficult

• Mixing up x-values and y-values: Writing y-coordinates in place of x-coordinates during substitution.
• Handling negative numbers: Getting confused when formulas contain built-in minus signs alongside negative point values.
• Subtracting negative numbers: Forgetting that subtracting a negative value turns into addition (e.g., 5 − (−2) = 5 + 2 = 7).
• Choosing correct coordinates for axes: Forgetting that any point on the x-axis has y = 0, and any point on the y-axis has x = 0.
• Understanding ratios in the section formula: Multiplying m₁ with x₁ instead of multiplying m₁ with x₂.
• Making calculation mistakes: Mistakes in squaring numbers or simplifying final square root values.

⚠️ Common Mistakes Students Make

1. Mixing x₁ with y₁ in formulas
   • Mistake: Writing (x₂ − y₁)² inside the distance formula instead of (x₂ − x₁)².
   • Correction: Always pair x-values together (x₂ − x₁) and y-values together (y₂ − y₁).
2. Forgetting brackets around negative numbers
   • Mistake: Writing −3² = −9 instead of (−3)² = 9.
   • Correction: Remember that the square of any real number (positive or negative) is always positive.
3. Changing the order of coordinates in the Section Formula
   • Mistake: Multiplying ratio m₁ with x₁ instead of x₂.
   • Correction: Cross-multiply the ratio! Multiplier m₁ goes with the second point (x₂), and m₂ goes with the first point (x₁).
4. Assuming points on axes incorrectly
   • Mistake: Assuming a point on the x-axis as (0, x) or a point on the y-axis as (y, 0).
   • Correction: A point on the x-axis is always (x, 0) and a point on the y-axis is always (0, y).
5. Using incorrect ratios when finding line divisions
   • Mistake: Using ratio 1 : 3 for trisection points instead of 1 : 2 and 2 : 1.
   • Correction: Trisection splits a line into 3 equal parts. The first point cuts it in ratio 1 : 2, and the second point cuts it in ratio 2 : 1.
6. Forgetting to simplify square roots
   • Mistake: Leaving √50 as the final answer instead of writing 5√2.
   • Correction: Always break down square roots into their simplest form (e.g., √50 = √(25 × 2) = 5√2).

🔥 Priority Revision

⭐⭐⭐ HIGH-PRIORITY PRACTICE

• Distance formula problems involving negative coordinates and square roots.
• Section formula problems to find the coordinates of a dividing point.
• Finding the ratio k : 1 when the dividing point or axis is given.

⭐⭐ IMPORTANT PRACTICE

• Finding the midpoint of a line segment.
• Finding the center of a circle when endpoints of a diameter are given.
• Proving if three points are collinear using distance formula.

⭐ ADDITIONAL REVISION

• Finding missing vertices of a parallelogram using the midpoint of diagonals property.
• Finding points on the x-axis or y-axis that are equidistant from two given points.

⚡ Coordinate Geometry Quick Revision Sheet

• Coordinates of a Point: Written as (x, y) where x is horizontal distance and y is vertical distance.
• Origin Coordinates: (0, 0).
• Distance Formula: AB = √[(x₂ − x₁)² + (y₂ − y₁)²].
• Distance from Origin: OP = √(x² + y²).
• Midpoint Formula: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
• Section Formula: P(x, y) = ((m₁x₂ + m₂x₁) ÷ (m₁ + m₂), (m₁y₂ + m₂y₁) ÷ (m₁ + m₂)).
• Important Warning: Negative squared is always positive! For example, (−5)² = 25.

📝 Test Yourself

1. Write the distance of the point P(−6, 8) from the origin (0, 0).
2. Find the distance between the pair of points A(−3, 4) and B(3, −4).
3. If the distance between the points (4, p) and (1, 0) is 5 units, calculate the value of p.
4. Find the coordinates of the midpoint of the line segment joining the points P(−5, 7) and Q(3, −1).
5. Find the coordinates of the point which divides the line segment joining A(2, −4) and B(7, 6) internally in the ratio 2 : 3.
6. In what ratio does the point P(2, 3) divide the line segment joining the points A(−1, 0) and B(4, 5)?
7. Find the coordinates of a point A, where AB is the diameter of a circle whose center is (3, −4) and B is the point (1, 2).
8. Find the point on the x-axis which is equidistant from the points P(3, 2) and Q(−5, 4).
9. Find the coordinates of the points of trisection of the line segment joining A(3, −3) and B(−6, 9).
10. Check whether the points A(1, 2), B(4, 5), and C(7, 8) lie on a single straight line (collinear).

🎯 Before Your Exam

☐ I understand coordinates and how to plot points correctly on a plane.

☐ I can identify the x-coordinate and y-coordinate of any given point.

☐ I can calculate the distance between any two points using the distance formula.

☐ I can find the midpoint of a line segment quickly and accurately.

☐ I understand how to use the section formula to find points and division ratios.

☐ I can simplify positive and negative numbers without sign mistakes.

☐ I know exactly which formula to pick based on what the question asks.

Stay confident, read each question carefully, and solve step by step!