Find the value of k so that the following equations may represent pairs of straight lines :
$x^{2} + \frac{1}{2} x y + y^{2} - 5 x - 7 y + k + 0$

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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 $x^{2} + \frac{1}{2} x y + y^{2} - 5 x - 7 y + k = 0$

$a = 1, b = 1, h = \frac{1}{4}, g = \frac{- 5}{2}, f = \frac{- 7}{2}, c = k Δ = (1 \times 1 \times k) + (2 \times \frac{- 7}{2} \times \frac{- 5}{2} \times \frac{1}{4}) - (1 \times \frac{- 7}{2} \times \frac{- 7}{2}) - (1 \times \frac{- 5}{2} \times \frac{- 5}{2}) - (k \times \frac{1}{4} \times \frac{1}{4}) = 0 k + \frac{35}{8} - \frac{49}{4} - \frac{25}{4} - \frac{k}{16} = 0 \frac{15 k}{16} - \frac{113}{8} = 0 k = \frac{113}{8} \times \frac{16}{15} = \frac{226}{15}$

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