DISTANCE FORMULA IN COORDINATE GEOMETRY

PRERNA FOR IAS

DISTANCE FORMULA IN COORDINATE GEOMETRY

1. Distance Formula in Coordinate Geometry

The Distance Formula is a fundamental concept in coordinate geometry used to calculate the distance between two points on a Cartesian plane. If two points are A(x₁, y₁) and B(x₂, y₂), the distance between them is given by:

AB = √[(x₂ − x₁)² + (y₂ − y₁)²]

This formula is derived from the Pythagorean Theorem and helps determine the length of a line segment joining two points. It is widely used in geometry, physics, engineering, and computer graphics. The distance formula allows accurate measurement of positions and relationships between points in two-dimensional space.

2. Distance of a Point from the Origin

The distance of a point from the origin is a special application of the distance formula. If a point P has coordinates (x, y), and the origin is O(0, 0), then the distance OP is:

OP = √(x² + y²)

This formula is obtained by substituting the origin's coordinates into the general distance formula. It helps determine how far a point lies from the center of the coordinate system. The concept is useful in geometry, navigation, and graph analysis. Understanding distance from the origin is important for studying circles, vectors, and various coordinate geometry problems.

3. Derivation of the Distance Formula Using Pythagoras Theorem

The distance formula is derived using the Pythagorean Theorem. Consider two points A(x₁, y₁) and B(x₂, y₂). By drawing horizontal and vertical lines, a right-angled triangle is formed. The horizontal side has length |x₂ − x₁|, and the vertical side has length |y₂ − y₁|. According to the Pythagorean Theorem:

AB² = (x₂ − x₁)² + (y₂ − y₁)²

Taking the square root of both sides gives:

AB = √[(x₂ − x₁)² + (y₂ − y₁)²]

This mathematical proof shows how geometry and algebra work together to calculate distances accurately.

4. Horizontal Distance

Horizontal distance refers to the distance between two points that lie on the same horizontal line. In this case, the y-coordinates are equal, meaning y₁ = y₂. Since there is no vertical change, the distance is simply the difference between the x-coordinates:

Distance = |x₂ − x₁|