If $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of parallel lines,then $\sqrt{\frac{g^2-ac}{f^2-bc}}$ is equal to

If the angle $2\theta$ is acute,then the acute angle between the lines represented by $x^2(\cos \theta - \sin \theta) + 2xy \cos \theta + y^2(\cos \theta + \sin \theta) = 0$ 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

The angle between the lines represented by the equation ${x^2} - xy - 6{y^2} - 7x + 31y - 18 = 0$ is.....$^o$

The combined equation of the two diameters of a circle which divide the circle into $4$ sectors is $ax^2+2hxy+by^2=0$. If the area of the bigger sector is $5$ times the area of the smaller sector,then $\frac{|a+b|}{\sqrt{(a-b)^2+4h^2}} = $

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} = $