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 $a+2 h+b=$

If the angle between the lines joining the origin to the points of intersection of $x+2 y+\lambda=0$ and $2 x^2-2 x y+3 y^2+2 x-y-1=0$ is $\frac{\pi}{2}$, then a value of $\lambda$. is

If $P$ is a variable point which is at a distance of 2 units. from the line $2 x-3 y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is