The lines $L_{1}$ and $L_{2}$ denoted by $3 x^{2} + 10 x y + 8 y^{2} + 14 x + 22 y + 15 = 0$ intersects at the point P and having slopes $m_{1}$ and $m_{2}$ respectively. The acute angle between them is $θ$ . Which of the following relations hold good

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$θ = \sin^{- 1} (\frac{1}{5 \sqrt{5}})$

$m_{1} m_{2} = \frac{2}{8}$

$m_{1} + m_{2} = \frac{5}{4}$

sum of abscissa and ordinate of point P is -1

answer is D.

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Detailed Solution

$a = 3, 2 h = 10, b = 8, 2 g = 14, 2 f = 22, c = 15$

$m_{1} + m_{2} = \frac{2 h}{b} = - \frac{5}{4}, m_{1} m_{2} = \frac{a}{b} = \frac{3}{8}$

$\sin θ = \frac{2 \sqrt{h^{2} - a b}}{\sqrt{{(a - b)}^{2} + 4 h^{2}}} = \frac{2}{5 \sqrt{5}}$

$P = (\frac{h f - b g}{a b - h^{2}}, \frac{g h - a f}{a b - h^{2}}) = (1, - 2)$

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