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

If $A$ is $(2, 5)$,$B$ is $(4, -11)$ and $C$ lies on $9x + 7y + 4 = 0$,then the locus of the centroid of the $\Delta ABC$ is a straight line parallel to the straight line:

If $A(1,0), B(0,-2), C(2,-1)$ are three fixed points,then the equation of the locus of a point $P(x,y)$ such that the area of $\triangle PAB$ is equal to the area of $\triangle PAC$ is:

The locus of $P$ such that the area of $\Delta PAB = 12 \text{ sq. units}$,where $A(2, 3)$ and $B(-4, 5)$ is:

The point $P$ is equidistant from $A(1, 3)$,$B(-3, 5)$,and $C(5, -1)$. Then $PA$ is equal to:

The locus of a point such that the sum of its distances from two given perpendicular lines is equal to $2$ units in the first quadrant is