What is the distance between the straight lines x - 2y = 4 and 2x - 4y = 10?

Concept:

Distance d between two parallel lines y = mx + c₁ and y = mx + c₂ is given by:

\(\rm d = \frac{|c_{1} - c_{2}|}{\sqrt{A^{2}+ B^{2}}}\)

Calculation:

L1: x - 2y = 4,     Slope of L1 = \(\rm \frac{-coefficient of x}{coefficient of y}\)\(\rm = \frac{-1}{-2} = \frac{1}{2}\)

L2: 2x - 4y = 10, Slope of L2 = \(\rm \frac{-coefficient of x}{coefficient of y}\)\(\rm = \frac{-2}{-4} = \frac{1}{2}\)

The slope is equal so lines are parallel to each other.

L2 can also be written as

L2: x - 2y = 5

Distance between parallel lines = \(\rm d = \frac{|c_{1} - c_{2}|}{\sqrt{A^{2}+ B^{2}}}\)

\(\rm d = \frac{|4 - 5|}{\sqrt{1^{2} + 2^{2}}}\)

\(\rm = \frac{1}{\sqrt{1 + 4}}\)

\(\rm = \frac{1}{\sqrt{5}}\)

Additional Information

Distance between two Straight Lines:

The distance between two straight lines in a plane is the minimum distance between any two points lying on the lines. In geometry, we often deal with different sets of lines such as parallel lines, intersecting lines, or skew lines.