IIT Madras
Mathematics 1
Distance formula

Distance formula

A well-defined collection of distinct objects called elements or members.

https://youtu.be/aDhyAkXiDOY

1️⃣ Distance of a Point from the Origin

To compute the distance of a point $P(x, y)$ from the origin $(0, 0)$ in the rectangular coordinate system, use the distance formula:

$$ OP = \sqrt{x^2 + y^2} $$

This formula comes straight from the Pythagorean Theorem, since the movement from $(0, 0)$ to $(x, y)$ forms the two right-angle legs.

Example: For the point $P(3, 4)$:

$$ OP = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Image:

Distance from Origin Example

source: cuemath.com The blue line shows the distance from the origin to a point $ (x, y) $.

2️⃣ General Distance Formula Between Two Points

The distance between any two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the coordinate plane is:

$$ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Derivation:

  • Draw a horizontal and vertical to create a right triangle:
    • One leg is $|x_2-x_1|$ (horizontal distance).
    • Other leg is $|y_2-y_1|$ (vertical distance).
  • By the Pythagorean theorem, the straight-line (hypotenuse) distance is as above.

Example: Find the distance between points $A(2, 1)$ and $B(7, 5)$:

$$ AB = \sqrt{(7 - 2)^2 + (5 - 1)^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} $$

Image:

Distance Between Two Points

source: byjus.com

Summary:

  • The distance from the origin to $ (x, y) $ uses $ \sqrt{x^2 + y^2} $.
  • The distance between any two points $ (x_1, y_1) $ & $ (x_2, y_2) $ is $ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $. 123456789
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Exercise Questions 🤯

Hello! On this Wednesday evening here in India, I would be happy to help you with these questions on the rectangular coordinate system. Let’s break them down one by one.

Question 1: Locating Points

The Question: Choose the correct option with respect to the points $P(5, -3)$, $Q(-3, 3)$, $R(0, -100)$, and $S(-2.5, 0)$ on the rectangular coordinate system.

  • Point R does not lie in any quadrant
  • Points P and R lie in Quadrant III
  • Points S and Q lie in Quadrant II
  • Points R and S cannot be represented on the rectangular coordinate system

Core Concepts: Quadrants and Axes

To solve this, we need to know where points are located based on the signs of their x and y coordinates.

  • Quadrant I: x is positive (+), y is positive (+)
  • Quadrant II: x is negative (-), y is positive (+)
  • Quadrant III: x is negative (-), y is negative (-)
  • Quadrant IV: x is positive (+), y is negative (-)
  • On an Axis: If either the x or y coordinate is 0, the point is not in a quadrant but lies on one of the axes.
    • If x = 0, the point is on the y-axis.
    • If y = 0, the point is on the x-axis.

Detailed Solution:

Let’s analyze the location of each point:

  1. P(5, -3): The x-coordinate (5) is positive, and the y-coordinate (-3) is negative. A (+, -) point lies in Quadrant IV.
  2. Q(-3, 3): The x-coordinate (-3) is negative, and the y-coordinate (3) is positive. A (-, +) point lies in Quadrant II.
  3. R(0, -100): The x-coordinate is 0. This means the point lies directly on the y-axis. Points on an axis are not in any quadrant.
  4. S(-2.5, 0): The y-coordinate is 0. This means the point lies directly on the x-axis. Points on an axis are not in any quadrant.

Now let’s evaluate the given options:

  • Point R does not lie in any quadrant: This is TRUE. As we determined, R(0, -100) lies on the y-axis.
  • Points P and R lie in Quadrant III: This is FALSE. P is in Quadrant IV, and R is on the y-axis.
  • Points S and Q lie in Quadrant II: This is FALSE. Q is in Quadrant II, but S is on the x-axis.
  • Points R and S cannot be represented on the rectangular coordinate system: This is FALSE. Any ordered pair of real numbers, including those with zero, can be precisely located on the plane.

Final Answer: The only correct option is “Point R does not lie in any quadrant”.

Question 2: Fundamentals of the Coordinate System

The Question: Which of the following is/are correct with respect to the rectangular coordinate system? (Multiple Select Question)

  • The horizontal line is called Y-axis
  • The point of intersection of the X and Y axes is called the origin
  • The vertical line is called X-axis
  • Any point on the coordinate plane can be represented as an ordered pair (x, y)

Core Concepts: Definitions

  • x-axis: The horizontal number line that passes through the origin.
  • y-axis: The vertical number line that passes through the origin.
  • Origin: The specific point $(0, 0)$ where the x-axis and y-axis intersect.
  • Ordered Pair: The standard notation $(x, y)$ that gives the unique “address” of any point by specifying its horizontal distance (x) and vertical distance (y) from the origin.

Detailed Solution:

Let’s check each statement against the definitions:

  1. The horizontal line is called Y-axis: This is FALSE. The horizontal line is the x-axis.
  2. The point of intersection of the X and Y axes is called the origin: This is TRUE. This is the definition of the origin.
  3. The vertical line is called X-axis: This is FALSE. The vertical line is the y-axis.
  4. Any point on the coordinate plane can be represented as an ordered pair (x, y): This is TRUE. This is the fundamental purpose of the coordinate system.

Final Answer: The two correct statements are “The point of intersection of the X and Y axes is called the origin” and “Any point on the coordinate plane can be represented as an ordered pair (x, y)”.

Question 3: Identifying Incorrect Representations

The Question: Identify the incorrect options for the representation of a point on the coordinate plane. (Multiple Select Question)

Core Concepts: Coordinate Conventions

This question tests the same concepts as Question 1 but asks you to find the mistakes. Let’s list the correct conventions first:

  • Quadrant I: (+, +)
  • Quadrant II: (-, +)
  • Quadrant III: (-, -)
  • Quadrant IV: (+, -)
  • On X-axis: (x, 0) or (±, 0)
  • On Y-axis: (0, y) or (0, ±)
  • Origin: (0, 0)

Detailed Solution:

Now we will evaluate each option to see if it’s correct or incorrect. The goal is to identify the incorrect ones.

  1. Quadrant I : (+, +): This statement is CORRECT.
  2. Quadrant IV : (-, -): This statement is INCORRECT. Quadrant IV points have a positive x and a negative y, so the form is (+, -).
  3. Quadrant II : (-, -): This statement is INCORRECT. Quadrant II points have a negative x and a positive y, so the form is (-, +).
  4. Quadrant III : (-, +): This statement is INCORRECT. Quadrant III points have a negative x and a negative y, so the form is (-, -).
  5. On X-axis: (0, ±): This statement is INCORRECT. For any point on the x-axis, the y-coordinate is always 0. The correct form is (±, 0).
  6. On Y-axis: (±, 0): This statement is INCORRECT. For any point on the y-axis, the x-coordinate is always 0. The correct form is (0, ±).
  7. Origin (0,0): This statement is CORRECT.

Final Answer: The incorrect options are:

  • Quadrant IV : (-, -)
  • Quadrant II : (-, -)
  • Quadrant III : (-, +)
  • On X-axis: (0, ±)
  • On Y-axis: (±, 0)

  1. https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(LibreTexts)/03:_Graphing_Lines/3.01:_Rectangular_Coordinate_System ↩︎

  2. https://byjus.com/jee/how-to-find-rectangular-coordinate-system-in-space/ ↩︎

  3. https://testbook.com/maths/how-to-find-rectangular-coordinate-system-in-space ↩︎

  4. https://en.wikipedia.org/wiki/Cartesian_coordinate_system ↩︎

  5. https://byjus.com/maths/distance-formula/ ↩︎

  6. https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs ↩︎

  7. https://www.askiitians.com/iit-jee-3d-geometry/rectangular-co-ordinate-system-in-space/ ↩︎

  8. https://nios.ac.in/media/documents/srsec311new/311_maths_eng/311_maths_eng_lesson13.pdf ↩︎

  9. https://www.cuemath.com/geometry/coordinate-geometry/ ↩︎

Rectangular Coordinate systemDistance of a line from a given point