Find the value of $a$ if the straight lines $5 x - 2 y - 9 = 0$ and $a y + 2 x - 11 = 0$ are perpendicular to each other.

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Solution

Consider the straight lines $5 x - 2 y - 9 = 0$ and $a y + 2 x - 11 = 0$

Given the lines are perpendicular, $m_{1} \times m_{2} = - 1$

Consider $5 x - 2 y - 9 = 0$

$\Rightarrow 2 y = 5 x - 9$

$\Rightarrow y = \frac{5}{2} x - \frac{9}{2}$

Slope $m_{1} = \frac{5}{2}$

Consider $a y + 2 x - 11 = 0$

$\Rightarrow a y = - 2 x + 11$

$\Rightarrow y = - \frac{2}{a} + \frac{11}{a}$

Slope $m_{2} = - \frac{2}{a}$

We have $m_{1} \times m_{2} = - 1$

$\Rightarrow \frac{5}{2} \times \frac{- 2}{a} = - 1$

$\Rightarrow \frac{5}{a} = 1$

$\Rightarrow a = 5$

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