$A$ point $P$ moves inside a square of area $4$ square units such that it is nearer to the point of intersection of its diagonals than any vertex. The area of the region traced by $P$ is

The equation of the locus of a point which is equidistant from the points $(2, 3)$ and $(4, 5)$ is

$A$ straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $OPRQ$ is a rectangle,then the locus of $R$ is

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

If $A (\cos \alpha, \sin \alpha)$,$B (\sin \alpha, -\cos \alpha)$,and $C (1, 2)$ are the vertices of $\Delta ABC$,find the locus of its centroid as $\alpha$ varies.

Through a given point $P(a, b)$,a straight line is drawn to meet the axes at $Q$ and $R$. If the parallelogram $OQSR$ is completed,then the equation of the locus of $S$ is (given $O$ is the origin):