If the equation $12x^2 + 7xy - py^2 - 18x + qy + 6 = 0$ represents a pair of perpendicular straight lines,then:

If the pair of straight lines given by $A x^2+2 H x y+B y^2=0$,where $(H^2>A B)$,forms an equilateral triangle with the line $a x+b y+c=0$,then $(A+3 B)(3 A+B)=$ (in $H^2$)

If the slopes of both the lines given by $x^2 + 2hxy + 6y^2 = 0$ are positive and the angle between these lines is $\operatorname{Tan}^{-1}\left(\frac{1}{7}\right)$,then find the value of $h$.

If $\theta$ is the angle between the lines represented by $x^2 - 3xy + \lambda y^2 + 3x - 5y + 2 = 0,$ where $\lambda$ is a real number,then $\csc^2 \theta$ equals:

The pair of lines $l x^2 + 2(l+m) x y + m y^2 = 0$ lies along two diameters of a circle and divides the circle into $4$ sectors. If the area of the bigger sector is $5$ times the area of the smaller sector,then $\frac{l m}{(l+m)^2} = $

If the angle between the lines given by the equation $x^{2}-3xy+\lambda y^{2}+3x-5y+2=0$,$\lambda \geq 0$,is $\tan^{-1}\left(\frac{1}{3}\right)$,then $\lambda=$