The lines $(lx + my)^2 - 3(mx - ly)^2 = 0$ and $lx + my + n = 0$ form

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} = $

Let $OABC$ be a parallelogram. The equation of one diagonal $AC$ is $x+y-1=0$ and the combined equation of the sides $OA, OC$ is $2x^2-y^2=0$. If $G$ is the centroid of the triangle $OAC$,then $BG=$

If $d$ is the distance between the point of intersection of the lines $x^2+4xy+ky^2-4x-10y+3=0$ and the origin,and $p$ is the product of the perpendicular distances from the origin to these lines,then $d^2-20p^2=$

If $P$ is the set of all real numbers $\alpha$ such that the product of the lengths of perpendiculars from $(\alpha, 1)$ to the pair of straight lines $3x^2+7xy+2y^2=0$ is $\frac{\sqrt{2}}{5}$,then the sum of the elements in $P$ is

If $(p, q)$ is the centroid of the triangle formed by the lines $8x^2 - 14xy + 5y^2 = 0$ and $x - 2y + 3 = 0$,then