Distance Between Two Parallel Lines

Understanding the distance between two parallel lines is a fundamental concept in coordinate geometry. Parallel lines are lines in a plane that never meet; they are always the same distance apart and do not intersect.

General Equation of a Line

Before we delve into the distance between parallel lines, let's recall the general equation of a line in the Cartesian plane:

$$ Ax + By + C = 0 $$

where $A$, $B$, and $C$ are constants.

Parallel Lines

Two lines are parallel if they have the same slope. For lines in the general form $Ax + By + C = 0$, this means that the lines have the same $A$ and $B$ coefficients but different $C$ coefficients.

For example, the lines:

$$ L_1: Ax + By + C_1 = 0 $$ $$ L_2: Ax + By + C_2 = 0 $$

are parallel if $C_1 \neq C_2$.

Distance Formula

The distance $d$ between two parallel lines $L_1$ and $L_2$ can be found using the following formula:

$$ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} $$

This formula is derived from the point-to-line distance formula and the fact that parallel lines have the same slope.

Table of Differences and Important Points

Property	Description
Slope	Parallel lines have identical slopes.
Intersecting	Parallel lines never intersect.
Distance	The distance between parallel lines is constant.
Coefficients	Parallel lines have the same $A$ and $B$ coefficients in their general equations.

Examples

Example 1: Basic Calculation

Given two parallel lines:

$$ L_1: 2x + 3y + 1 = 0 $$ $$ L_2: 2x + 3y - 4 = 0 $$

Find the distance between them.

Solution:

Here, $A = 2$, $B = 3$, $C_1 = 1$, and $C_2 = -4$. Using the distance formula:

$$ d = \frac{|-4 - 1|}{\sqrt{2^2 + 3^2}} = \frac{5}{\sqrt{13}} \approx 1.39 $$

Example 2: Horizontal and Vertical Lines

Consider two horizontal lines:

$$ L_1: y = 5 $$ $$ L_2: y = -3 $$

Find the distance between them.

Solution:

For horizontal lines, the general form is $0x + 1y + C = 0$. Thus, $A = 0$, $B = 1$, $C_1 = -5$, and $C_2 = 3$. Using the distance formula:

$$ d = \frac{|3 - (-5)|}{\sqrt{0^2 + 1^2}} = 8 $$

The distance between the two horizontal lines is 8 units.

Example 3: Application in Geometry

Two parallel lines are part of a rectangle's sides. If the length of the rectangle is 10 units and the width is 5 units, what is the distance between the longer sides?

Solution:

The distance between the longer sides of the rectangle is equal to the width of the rectangle, which is 5 units.

In conclusion, the distance between two parallel lines is a straightforward calculation once you have the lines in the general form. It's a useful concept in various fields, including geometry, physics, and engineering.