If $A=(5,3)$,$B=(3,-2)$ and a point $P$ is such that the area of the triangle $PAB$ is $9$,then the locus of $P$ represents

The coordinates of the points $O$,$A$,and $B$ are $(0,0)$,$(0,4)$,and $(6,0)$ respectively. If a point $P$ moves such that the area of $\Delta POA$ is always twice the area of $\Delta POB$,then the equation to both parts of the locus of $P$ is

The equation $x^{3}-y x^{2}+x-y=0$ represents

$A$ variable line through the point $P(-1, 2)$ cuts the coordinate axes at $A$ and $B$ respectively. If $Q$ is a point on $AB$ such that $PA, PQ, PB$ are in a harmonic progression,then the locus of $Q$ is

$A$ variable line $\frac{x}{a}+\frac{y}{b}=1$ is such that $a+b=4$. The locus of the midpoint of the portion of the line intercepted between the axes is

Given $A(1, 1)$ and $AB$ is any line through it cutting the $x-$ axis in $B$. If $AC$ is perpendicular to $AB$ and meets the $y-$ axis in $C$,then the equation of the locus of the midpoint $P$ of $BC$ is