Let $A = (0, 4)$ and $B = (2 \cos \theta, 2 \sin \theta)$,for some $0 < \theta < \frac{\pi}{2}$. Let $P$ divide the line segment $AB$ in the ratio $2:3$ internally. The locus of $P$ 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 $,\%$)?

$A$ 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

The locus of the centre of a circle touching both the coordinate axes is

At which point on the line $x = 3$ are the tangents drawn to the circle $x^2 + y^2 = 8$ perpendicular to each other?

The locus of the midpoints of the chords of the circle $C_1: (x-4)^2 + (y-5)^2 = 4$ which subtend an angle $\theta_i$ at the centre of the circle $C_1$,is a circle of radius $r_i$. If $\theta_1 = \frac{\pi}{3}$,$\theta_3 = \frac{2\pi}{3}$ and $r_1^2 = r_2^2 + r_3^2$,then $\theta_2$ is equal to