Can you please derive the distance between two parallel lines' formula - Maths - Straight Lines

Question

Can you please derive the distance between two parallel lines'
formula ??

Lalit Mehra · Accepted Answer

Let ax + by + c1 = 0 and ax + by + c2 = 0 be two parallel lines Suppose P(x1, y1) is a point on the line ax + by + c1 = 0 âˆ´ ax1 + by1 + c1 = 0 â‡’ ax1 + by1 = â€“ c1 (1) Distance between two parallel lines, PQ = Length of perpendicular from P(x1, y1) on line ax + by + c2 = 0 $= | \frac{a x_{1} + b y_{1} + c_{2}}{\sqrt{a^{2} + b^{2}}} | [\begin{array}{l} \begin{matrix} Length of perpendicular from point (x_{1}, y_{1}) to the line \end{matrix} \\ a x + b y + c = 0 is | \frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}} | \end{array}] = | \frac{- c_{1} + c_{2}}{\sqrt{a^{2} + b^{2}}} | [U sing (1)] = | \frac{c_{2} - c_{1}}{\sqrt{a^{2} + b^{2}}} |$ âˆ´ Distance between the given parallel lines $= | \frac{c_{2} - c_{1}}{\sqrt{a^{2} + b^{2}}} |$

Can you please derive the distance between two parallel lines' formula ??