Mastering Coordinate Geometry: The Foundation

Welcome to the foundational module of Coordinate Geometry for JEE Mathematics. This section will equip you with the essential tools to navigate and analyze geometric shapes on a 2D plane. We'll start with the very basics: the coordinate system and how to measure distances between points.

The Cartesian Coordinate System

The Cartesian coordinate system, named after René Descartes, provides a framework for locating any point in a plane using two perpendicular lines called axes. The horizontal line is the x-axis, and the vertical line is the y-axis. Their intersection is the origin (0,0). Any point on the plane is represented by an ordered pair (x, y), where 'x' is the horizontal distance from the origin (abscissa) and 'y' is the vertical distance from the origin (ordinate).

Every point on a plane has a unique address.

The coordinate system uses two perpendicular axes (x and y) to define the location of any point as an ordered pair (x, y). The first number (x) tells you how far to move horizontally from the origin, and the second number (y) tells you how far to move vertically.

The plane is divided into four quadrants by the x and y axes. Quadrant I (x>0, y>0), Quadrant II (x<0, y>0), Quadrant III (x<0, y<0), and Quadrant IV (x>0, y<0). Understanding these quadrants is crucial for visualizing the position of points and the orientation of geometric figures.

The Distance Formula

The distance formula is a direct application of the Pythagorean theorem. It allows us to calculate the straight-line distance between two points in a Cartesian plane. If we have two points, P1 with coordinates (x1, y1) and P2 with coordinates (x2, y2), the distance 'd' between them is given by the formula:

The distance formula is derived from the Pythagorean theorem ( $a^2 + b^2 = c^2$ ). Consider the two points $(x_1, y_1)$ and $(x_2, y_2)$ . We can form a right-angled triangle where the horizontal leg has length $|x_2 - x_1|$ and the vertical leg has length $|y_2 - y_1|$ . The distance between the two points is the hypotenuse of this triangle. Therefore, $d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$ . Taking the square root of both sides gives us the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . This formula is fundamental for many geometric calculations.

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The formula is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Remember: The order of the points in the formula doesn't matter because the differences are squared, making the result always positive.

What is the distance between the points (2, 3) and (5, 7)?

Using the distance formula: $d = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ units.

Applications in JEE Mathematics

Understanding the coordinate system and the distance formula is paramount for tackling more complex topics in JEE Mathematics, including:

Finding the length of line segments.
Determining if points are collinear.
Calculating the perimeter and area of polygons.
Identifying properties of geometric shapes like triangles and quadrilaterals.
Laying the groundwork for understanding equations of lines, circles, and other conic sections.

If the distance between (a, 2) and (3, 4) is 2, what are the possible values of 'a'?

Using the distance formula: $2 = \sqrt{(3-a)^2 + (4-2)^2} \Rightarrow 4 = (3-a)^2 + 2^2 \Rightarrow 4 = (3-a)^2 + 4 \Rightarrow (3-a)^2 = 0 \Rightarrow 3-a = 0 \Rightarrow a = 3$ .

Learning Resources

Khan Academy: The Distance Formula(video)

A clear video explanation of the distance formula and its derivation from the Pythagorean theorem.

Math is Fun: Coordinate Geometry(documentation)

An introductory overview of coordinate geometry, covering the Cartesian plane, points, and basic distance concepts.

Brilliant.org: Distance Formula(blog)

Explains the distance formula with interactive examples and visual aids, focusing on conceptual understanding.

Byju's: Coordinate Geometry - Distance Formula(documentation)

Provides a detailed explanation of the distance formula, including its proof and solved examples relevant to competitive exams.

NCERT Mathematics - Class 10 (Chapter 7: Coordinate Geometry)(documentation)

The official textbook chapter covering the coordinate system and the distance formula, essential for foundational understanding.

Vedantu: Distance Formula - Concepts and Examples(blog)

Offers a comprehensive guide to the distance formula with various types of problems and solutions.

Toppr: Coordinate Geometry - Distance Formula(blog)

A focused explanation of the distance formula, including its derivation and application in solving geometry problems.

Art of Problem Solving: Coordinate Geometry Introduction(documentation)

A more advanced resource that introduces coordinate geometry concepts, including the distance formula, within a problem-solving context.

YouTube: Distance Formula Explained (JEE Level)(video)

A video tutorial specifically tailored for competitive exams, demonstrating how to apply the distance formula to JEE-style problems.

Wikipedia: Cartesian Coordinate System(wikipedia)

A detailed historical and mathematical overview of the Cartesian coordinate system and its fundamental principles.

Coordinate System and Distance Formula