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The equation $x^{2} - 5 x y + p y^{2} + 3 x - 8 y + 2 = 0$ represents a pair of straight lines. If $θ$ is the angle between them, then $\sin θ =$

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$\frac{1}{\sqrt{50}}$

$\frac{1}{7}$

$\frac{1}{5}$

$\frac{1}{\sqrt{10}}$

answer is A.

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Detailed Solution

$a = 1, 2 h = - 5, b = p, 2 g = 3, 2 f = - 8, c = 2$

$a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0 \Rightarrow p = 6$

$\Rightarrow \sin θ = \frac{2 \sqrt{h^{2} - a b}}{\sqrt{{(a - b)}^{2} + 4 h^{2}}} = \frac{1}{\sqrt{50}}$

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