Show that the following equations represent a pair of lines, If yes find the angle between them
(i) $2 x^{2} + x y - 3 y^{2} - 6 x + 19 y - 20 = 0$

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Solution

$\begin{matrix} 2 x^{2} + x y - 3 y^{2} - 6 x + 19 y - 20 = 0 a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + x = 0 a = 2, h = \frac{1}{2}, b = - 3, g = - 3, f = \frac{19}{2} \end{matrix}$
If represent a pair of straight line, then
$\begin{matrix} a b c + f g h - a f^{2} - b g^{2} - c h^{2} = 0 2 \times (- 3) \times (- 20) + 2 \times \frac{19}{2} (- 3) (\frac{1}{2}) - 2 \times \frac{361}{4} \frac{120}{1} - \frac{57}{2} - \frac{361}{2} + \frac{27}{1} + \frac{10}{2} \frac{240 - 57 - 361 + 54 + 10}{2} \neq 0 \end{matrix}$
does not represent.

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