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

Solution :
We have, $$2{x^2} + 5xy + 3{y^2} + 6x + 7y + 4 = 0$$
Comparing this equation with $$a{x^2} + b{y^2} + 2hxy + 2gx + 2fy + c = 0,$$         we get $$a = 2,\,b = 3,\,h = \frac{5}{2}$$
$$\eqalign{ & \therefore \,\tan \,\theta = \frac{{2\sqrt {{h^2} - ab} }}{{a + b}} \cr & = \frac{{2\sqrt {\frac{{25}}{4} - 2 \times 3} }}{{2 + 3}} \cr & = \frac{{2\sqrt {\frac{1}{4}} }}{5} \cr & = \frac{{2 \times \frac{1}{2}}}{5} \cr & = \frac{1}{5}\tan \,\theta = \frac{1}{5} \cr & \Rightarrow m = \frac{1}{5} \cr} $$