Let $A\ (2, -3)$ and $B\ (-2, 1)$ be the vertices of a triangle $ABC$. If the centroid of this triangle moves on the line $2x + 3y = 1$,then the locus of the vertex $C$ is which line?

Suppose we have to cover the $XY$-plane with identical tiles such that no two tiles overlap and no gap is left between the tiles. Suppose that we can choose tiles of the following shapes: equilateral triangle,square,regular pentagon,regular hexagon. Then,the tiling can be done with tiles of

$A$ line is at a constant distance $c$ from the origin and meets the coordinate axes in $A$ and $B$. The locus of the centre of the circle passing through $O, A, B$ is

$A$ man starts walking from the point $P(-3, 4)$,touches the $x$-axis at $R$,and then turns to reach the point $Q(0, 2)$. The man is walking at a constant speed. If the man reaches the point $Q$ in the minimum time,then $50((PR)^{2} + (RQ)^{2})$ is equal to ..... .

Let $S$ be the set of points whose abscissae and ordinates are natural numbers. Let $P \in S$ be such that the sum of the distances of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then,the number of such points $P$ in $S$ is

If a variable straight line passing through the point of intersection of the lines $x-2y+3=0$ and $2x-y-1=0$ intersects the $X, Y$-axes at $A$ and $B$ respectively,then the equation of the locus of a point which divides the segment $AB$ in the ratio $-2:3$ is