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What is the distance between the straight lines x - 2y = 4 and 2x - 4y = 10?
1. 1/5
2. 1/√5 
3. 1
4. None of these

1 Answer

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Correct Answer - Option 2 : 1/√5 

Concept:

Distance d between two parallel lines y = mx + c1 and y = mx + cis 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 = \(\rm d = \frac{|c_{1} - c_{2}|}{\sqrt{A^{2}+ B^{2}}}\)

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

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

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

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.

Distance between Two Parallel Lines The perpendicular distance from any point on one line to the other line.
Distance between Two Intersecting Lines The shortest distance between such lines is eventually zero.
Distance between Two Skew Lines The distance is equal to the length of the perpendicular between the lines.

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