The measure of the angle between the lines $x^{2}+2xy \operatorname{cosec} \alpha+y^{2}=0$ is

If the lines represented by $(1+\sin^2 \theta) x^2+2hxy+2\sin \theta y^2=0$,where $\theta \in [0, 2\pi]$,are perpendicular to each other,then $\theta = \dots$.

The angle between the pair of lines $2x^2 + 5xy + 2y^2 + 3x + 3y + 1 = 0$ is

The measure of the acute angle $\theta$ between the pair of lines represented by the equation $2x^2 + xy - y^2 - x + 2y - 1 = 0$ is:

If the angle between the pair of straight lines represented by the equation ${x^2} - 3xy + \lambda {y^2} + 3x - 5y + 2 = 0$ is ${\tan ^{ - 1}}\left( {\frac{1}{3}} \right)$,where $\lambda$ is a non-negative real number,then $\lambda$ is:

If $\theta$ is the acute angle between the lines represented by the equation $2x^2 + 7xy + 3y^2 = 0$, then the value of $\frac{2 \cos \theta - 3 \sin \theta}{4 \sin \theta + 5 \cos \theta}$ is: