Distance Between Parallel Lines
Distance Between Parallel Lines
Introduction:
 The distance between parallel lines refers to the shortest distance measured perpendicular from any point on one line to the other parallel line.
Formula for Distance between Parallel Lines:
 Given two parallel lines:
 Line 1: $Ax+By+{C}_{1}=0$
 Line 2: $Ax+By+{C}_{2}=0$
 The distance 'd' between these parallel lines is given by:
$$\text{Distance}=\frac{\mathrm{\mid}{C}_{2}{C}_{1}\mathrm{\mid}}{\sqrt{{A}^{2}+{B}^{2}}}$$
Steps to Find Distance:
 Calculate Numerator:
 Subtract the constants (intercepts) of the equations of the parallel lines: $\mathrm{\mid}{C}_{2}{C}_{1}\mathrm{\mid}$.
 Calculate Denominator:
 Use the coefficients of $x$ and $y$ in the line equations to find $\sqrt{{A}^{2}+{B}^{2}}$
 .
 Divide Numerator by Denominator:
 Divide $\mathrm{\mid}{C}_{2}{C}_{1}\mathrm{\mid}$ by $\sqrt{{A}^{2}+{B}^{2}}$
 to determine the distance.
Example:
Given two parallel lines:

Line 1: $3x+4y5=0$

Line 2: $3x+4y+7=0$

Calculate Numerator:
 $\mathrm{\mid}{C}_{2}{C}_{1}\mathrm{\mid}=\mathrm{\mid}7(5)\mathrm{\mid}=\mathrm{\mid}12\mathrm{\mid}$

Calculate Denominator:
 Use coefficients of $x$ and $y$: $\sqrt{{3}^{2}+{4}^{2}}=\sqrt{9+16}=\sqrt{25}=5$

Find Distance:
 Distance $=\frac{\mathrm{\mid}12\mathrm{\mid}}{5}=\frac{12}{5}=2.4$ units
Importance:
 Determines the shortest distance between two parallel lines.
 Crucial in geometry, architecture, and engineering for accurate measurements and spatial planning.
Summary:
 The distance between parallel lines is computed using the difference in their constants divided by the square root of the sum of the squares of their coefficients.
 This measurement provides critical information for spatial analysis, optimization, and design in various fields.