If the equations $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ represent two perpendicular lines then prove that $a_{1} a_{2} + b_{1} b_{2} = 0$

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Solution

if both the lines are perpendicular then, $m_{1} m_{2} = - 1$
$m_{1}, m_{2}$ are slopes of lines,

$m_{1} = \frac{- b_{1}}{a_{1}}$

$m_{2} = \frac{- b_{2}}{a_{2}}$

$m_{1} m_{2} = - 1$

$\frac{b_{1} b_{2}}{a_{1} a_{2}} = - 1$

$b_{1} b_{2} = - a_{1} a_{2}$
$a_{1} a_{2} + b_{1} b_{2} = 0$

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