The pair of straight lines represented by the equation $\sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = 0$ are

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:

If $\varphi$ is the angle between the lines $a x^{2}+2 h x y+b y^{2}=0$,then the angle between the lines $x^{2}+2 x y \sec \theta+y^{2}=0$ is

If the pair of straight lines $xy-x-y+1=0$ and the line $x+ay-3=0$ are concurrent,then the acute angle between the pair of lines $ax^2-13xy-7y^2+x+23y-6=0$ is

If the acute angle between the lines given by $ax^2+2hxy+by^2=0$ is $\frac{\pi}{4}$,then $4h^2=$

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: