Prove that the straight lines $x + 2 y + 1 = 0$ and $2 x - y + 5 = 0$ are perpendicular to each other.

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Solution

Slope of the straight line $x + 2 y + 1 = 0$ is $m_{1} = - \frac{c o e f f i c i e n t o f x}{c o e f f i c i e n t o f y} = - \frac{1}{2}$ \
Slope of the straight line $2 x - y + 5 = 0$ is $m_{2} = - \frac{c o e f f i c i e n t o f x}{c o e f f i c i e n t o f y} = - \frac{- 2}{- 1} = 2$
Product of the slopes $m_{1} m_{2} = - \frac{1}{2} \times 2 = - 1$
$∴$ The two straight lines are perpendicular.

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