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

Let $A (2, 3)$ and $B (-4, 5)$ be two fixed points. $A$ point $P$ moves in such a way that the area of $\Delta PAB = 12 \, \text{sq. units}$. Find its locus.

Let the algebraic sum of the perpendicular distances from the points $A(2, 0)$,$B(0, 2)$,and $C(1, 1)$ to a variable line be zero. Then,all such lines:

The perpendicular bisector of the line segment joining the points $P(1, 4)$ and $Q(k, 3)$ has a $y$-intercept of $-4$. Then a possible value of $k$ among the following is:

$A$ point $P$ moves on the line $2x - 3y + 4 = 0$. If $Q(1, 4)$ and $R(3, -2)$ are fixed points,then the locus of the centroid of $\Delta PQR$ is a line

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