If one end of the diameter is $(1, 1)$ and the other end lies on the line $x+y=3$,then the locus of the centre of the circle 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 $,\%$)?

If the distance from a variable point $P$ to a fixed point $A(a, 0)$ is equal to the perpendicular distance from $P$ to the line $x+y=0$,then the equation of the locus of $P$ is

The locus of the centre of the circle $\frac{1}{2} (x^2 + y^2) + x \cos \theta + y \sin \theta - 4 = 0$ is :-

The centroid of a variable triangle $ABC$ is at a distance of $5$ units from the origin. If $A = (2, 3)$ and $B = (3, 2)$,then the locus of $C$ is

The locus of the point whose ratio of distance from the origin to its distance from $(-2, -3)$ is $5: 7$,is given by: