Find the value of k, if the following equations represent a pair of lines $3 x^{2} - 10 x y + 3 y^{2} + 16 y + k = 0$

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Solution

compare the equation with $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
$a = 3$ , $b = 3$ , $h = - 5$ , $g = 0$ , $f = 8$ , $c = k$
if $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$ then this equation represents a pair of straight lines
$9 k - 192 - 25 k = 0$
$k = - 12$

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