The difference of the tangents of the angles which the lines $(\tan ^2 \alpha+\cos ^2 \alpha) x^2-2 x y \tan \alpha +(\sin ^2 \alpha) y^2=0$ make with the $X$-axis is

If the lines $x^2-4xy+y^2=0$ and $x+y=10$ contain the sides of an equilateral triangle,then the area of the equilateral triangle is

From the point $(3,-4)$,perpendicular lines $L_1$ and $L_2$ are drawn to each of the lines represented by $S \equiv 2x^2+3xy-2y^2-7x+y+3=0$. The area of the quadrilateral formed by the pair of lines $S=0$,$L_1$,and $L_2$ is (in square units):

The combined equation of the diagonals of the square formed by the two pairs of straight lines given by $xy+4x-3y-12=0$ and $xy-3x+4y-12=0$ is

If $9x^2-24xy+16y^2+\alpha x+\beta y+6=0$ represents a pair of parallel lines $1$ unit apart and one of those lines passes through $(1,1)$,then $\frac{\alpha}{\beta} = $

If the sides of a triangle $ABC$ are $2x^2-y^2=0$ and $x+y-1=0$,and the sides of another triangle $PQR$ are $2x^2-5xy+2y^2=0$ and $7x-2y-12=0$,then the distance between the centroid of $\triangle ABC$ and the orthocentre of $\triangle PQR$ is