A straight line passing through a point $(3,2)$ cuts $X$ and $Y$ axes at the points $A$ and $B$ respectively. If a point $P$ divides $A B$ in the ratio $2: 3$, then the equation of the locus of point $P$ is

By shifting the origin to the point $(-1,2)$ through translation of axes, if $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ is the transformed equation of $2 x^2-x y+y^2-3 x+4 y-5=0$, then $2(f+g+h)=$

If a line $L$ passing through the point $A(-2,4)$ makes an angel of $60^{\circ}$ with the positive direction of $X$ - axis in anti-clockwise direction and $B(p, q)$ lying in the 3rd quadrant is a point on $L$ at the distance of 6 units from the point $A$, then $\sqrt{p^2+q^2-8 q}=$

If the perpendicular drawn from the point $(2,-3)$ to the straight line $4 x-3 y+8=0$ meets it at $M(a, b)$ and $a^3-b^3=k^3$, then $k=$