A straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $O P R Q$ is a rectangle, then the locus of $R$ is

If the lines $x+2 a y+a=0, x+3 b y+b=0$, $x+4 c y+c=0$ are concurrent, then $a, b, c$ are in

If $M$ is the foot of the perpendicular drawn from the origin to the line $x-2 y+3=0$ which meets the $X$ and $Y$-axes at $A$ and $B$, respectively, then $A M=$

One line of the pair of lines $x^2+x y-2 y^2=0$ is perpendicular to one line of the pair of lines $3 y^2-5 x y-2 x^2=0$ If the combined equation of the two lines other than those two perpendicular lines is $a x^2+2 h x y+b y^2=0$, then $a+2 h+b=$