Calculate the perpendicular distance between the parallel lines $2x-y+3=0$ and $2x-y-5=0$.

Step-by-step solution

1. Step 1: Identify the coefficients: for both lines $a=2$, $b=-1$, $c_1=3$, $c_2=-5$.
2. Step 2: Use the distance formula for parallel lines: $d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$.
3. Step 3: Compute the numerator: $|3-(-5)|=|8|=8$.
4. Step 4: Compute the denominator: $\sqrt{2^2+(-1{)}^2}=\sqrt{4+1}=\sqrt{5}$.
5. Step 5: Divide: $d=\frac{8}{\sqrt{5}}$.
6. Step 6: The perpendicular distance between the two lines is $\frac{8}{\sqrt{5}}$ (or $\frac{8\sqrt{5}}{5}$ if rationalised).

Concept tested: Distance between parallel lines

Published 4 October 2026 · How this solution was prepared and checked