$A$ point moves in such a way that its distance from $(1, -2)$ is always twice its distance from $(-3, 5)$. The locus of the point is:

Find the locus of the midpoint of the chord of the circle $x^2 + y^2 = 16$ which is a tangent to the hyperbola $9x^2 - 16y^2 = 144$.

The locus of the centre of the circle which cuts off intercepts of length $2a$ and $2b$ from the $x$-axis and $y$-axis respectively,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 a point $P$ moves such that the sum of the distances from $P$ to the points $A(1, -1)$ and $B(-1, 1)$ is always $4$,then the equation for the locus of $P$ is

$A$ variable circle is described to pass through the point $(1, 0)$ and is tangent to the curve $y = \tan(\tan^{-1} x)$. The locus of the centre of the circle is a parabola whose :