Find the value of k so that the following equations may represent pairs of straight lines :
$k x y - 8 x + 9 y - 12 = 0 .$

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Solution

The general equation of second degree $a x^{2} + 2 h x y + {b y}^{2} + 2 g x + 2 f y + c = 0$ represents a pair of straight lines if $Δ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$

For $k x y - 8 x + 9 y - 12 = 0$

$a = 0, b = 0, h = \frac{k}{2}, g = - 4, f = \frac{9}{2}, c = - 12 Δ = (0 \times 0 \times - 12) + (2 \times \frac{9}{2} \times - 4 \times \frac{k}{2}) - (0 \times \frac{9}{2} \times \frac{9}{2}) - (0 \times - 4 \times - 4) - (- 12 \times \frac{k}{2} \times \frac{k}{2}) = 0 0 - 18 k - 0 - 0 + 3 k^{2} = 0 3 k^{2} - 18 k = 0 3 k (k - 6) = 0 \Rightarrow k = 0, 6$

$k = 0$ is not possible as our equation reduces to equation of straight line when $k = 0$
$∴ k = 6$

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