Let $C$ be a curvea $x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ in a cartesian plane. By rotating the coordinate axes through an angle $\frac{\pi}{4}$ in the positive direction, if the transformed equation of $C$ is $Y^2+X Y-X=0$, then $\left(h^2-a b\right)-2 g f=$

If the straight line passing through the point $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive direction of $X$-axis and meets the line $12 x+5 y+10=0$ at $Q$, then the length of $P Q$ is

If the equation of the straight line passing through the point of intersection of $x+2 y-19=0, x-2 y-3=0$ and which is at a perpendicular distance of 5 units from the point $(-2,4)$ is $5 x+b y+c=0$, then a possible value of $5+b+c$ is

Suppose $O(0,0)$ is the origin and the line $L=x+y-\lambda=0$ meets the curve $x^2+y^2-2 x-4 y+2=0$ at $A$ and $B$. If $\angle A O B=90^{\circ}$, then the distance between such lines $L=0$ is