Let the locus of the point of intersection of the perpendicular tangents drawn to the circle $x^2+y^2+6x-4y-12=0$ be the circle $S$. Then the equation of the tangent drawn to $S$ which is perpendicular to the line $6x-4y+k=0$ 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 $,\%$)?

For all values of $\theta$,the locus of the point of intersection of the lines $x \cos \theta + y \sin \theta = a$ and $x \sin \theta - y \cos \theta = b$ is

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

$A$ point $P(x, y)$ is such that its distances from $(-1, 0)$ and $(0, 2)$ are in the ratio $\sqrt{2} : 1$. Then the locus of $P$ is