If the angle between the lines represented by $2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 4 = 0$ is ${tan}^{- 1} (m)$ then find $m$ .

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Solution

$\begin{matrix} C o m p a r e w i t h t h e e q u a t i o n, a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 r y + c = 0 a = 2, h = \frac{5}{2}, b = 3 ∴ tan Q = \frac{2 \sqrt{h^{2} - a b}}{| a + b |} = \frac{2 \sqrt{\frac{25}{4} - 6}}{5} = \frac{2 \times \frac{1}{2}}{5} tan Q = \frac{1}{5} Q = {tan}^{- 1} (\frac{1}{5}) ∴ m = \frac{1}{5} a n d 2 a + 3 b = 2 \times 2 + 3 \times 3 = 4 + 9 = 13 \end{matrix}$

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2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 4 = 0

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If the angle between the two lines represented by $2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 4 = 0$ is ${tan}^{- 1} m$ , then m =

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