The length of $x$-intercept made by pair of lines $2 x^2+x y-6 y^2-2 x+17 y-12=0$ is

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $x y$ form from the equation $3 x^2+2 \sqrt{3} x y+y^2=0$. Then, in the new coordinate system the equation $x^2+y^2+2 x y=2$ is transformed to

$P$ is a point on $x+y+5=0$, whose perpendicular distance from $2 x+3 y+3=0$ is $\sqrt{13}$, then the coordinates of $P$ are

For $\lambda, \mu \in R,(x-2 y-1)+\lambda(3 x+2 y-11)=0$ and $(3 x+4 y-11)+\mu(-x+2 y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $a x+b y-5=0$. Then, $2 a+b=$