Find the value of $k$ so that the line $k x - k y - 1 = 0$ may be perpendicular to $2 x - k y - 2 = 0$

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Solution

Given: $k x - k y - 1 = 0$ ....... $(1)$
Slope of line $(1) = - \frac{c o e f f i c i e n t o f x}{c o e f f i c i e n t o f y} = - \frac{k}{- k} = 1$
Given: $2 x - k y - 2 = 0$ ....... $(2)$
Slope of line $(2) = - \frac{c o e f f i c i e n t o f x}{c o e f f i c i e n t o f y} = - \frac{2}{- k} = \frac{2}{k}$
As the lines $(1)$ and $(2)$ are perpendicular, the product of their slopes $= - 1$
$\Rightarrow 1 \times \frac{2}{k} = - 1$
$\Rightarrow 2 = - k$
$∴ k = - 2$

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