$A$ rod of length $6$ units slides with its ends on the coordinate axes. The locus of the midpoint of the rod is

$A$ company situated at $(2,0)$ in the $XY$-plane charges $RS. 2$ per $km$ for delivery. $A$ second company at $(0,3)$ charges $RS. 3$ per $km$ for delivery. The region of the plane where it is cheaper to use the first company is

The locus of the centre of circles passing through $(a, b)$ and cutting the circle $x^2+y^2-2x+4y-4=0$ orthogonally is

$A$ solid hemisphere is attached to the top of a cylinder,having the same radius as that of the cylinder. If the height of the cylinder were doubled (keeping both radii fixed),the volume of the entire system would have increased by $50\,\%$. By what percentage would the volume have increased if the radii of the hemisphere and the cylinder were doubled (keeping the height fixed) (in $,\%$)?

$A$ circle passing through the origin cuts the coordinate axes at $A$ and $B$. If the straight line $AB$ passes through a fixed point $(x_1, y_1)$,then the locus of the centre of the circle is:

$A$ line segment $AM = a$ moves in the $XOY$ plane such that $AM$ is parallel to the $X$-axis. If $A$ moves along the circle $x^2 + y^2 = a^2$,then the locus of $M$ is