Find the distance between the parallel lines $3 x + 4 y + 7 = 0$ and $3 x + 4 y - 5 = 0$ from the origin.

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Solution

Solve for distance between the parallel lines from origin
Compare the given lines with $a x + b y + c_{1} = 0$ and $a x + b y + c_{2} = 0$
$\Rightarrow a = 3, b = 4$
$\Rightarrow c_{1} = 7, c_{2} = - 5$
Distance between two parallel lines from origin is
$d = \frac{|c_{1} - c_{2}|}{\sqrt{a^{2} + b^{2}}}$
$∴ d = \frac{|7 - (- 5)|}{\sqrt{3^{2} + 4^{2}}} = \frac{|12|}{\sqrt{25}} = \frac{12}{5}$
Hence, distance between the given parallel lines from origin is $\frac{12}{5}$

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