From the origin,chords are drawn to the circle $(x - 1)^2 + y^2 = 1$. The locus of the midpoints of these chords is a

$A$ stick of length $10$ units rests against the floor and a wall of a room. If the stick begins to slide on the floor,then the locus of its middle point is:

If the ratio of the distances of a variable point $P$ from the point $(1, 1)$ and the line $x-y+2=0$ is $1: \sqrt{2}$,then the equation of the locus of $P$ is

$A$ variable chord is drawn through the origin to the circle $x^2 + y^2 - 2ax = 0$. The locus of the centre of the circle drawn on this chord as diameter is:

If a circle passing through $A(1,1)$ touches the $X$-axis,then the locus of the other end of the diameter through $A$ is

Let $A$ be the point $(3, 0)$ and circles with variable diameter $AB$ touch the circle $x^2 + y^2 = 36$ internally. Let the curve $C$ be the locus of the point $B$. If the eccentricity of $C$ is $e$, then $72e^2$ is equal to . . . . . .