$A$ stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is

Let $P(\alpha, \beta)$ be a variable point which moves in the $x-y$ plane such that $\frac{PA}{PB} = 2$,where $A(1, 0)$ and $B(0, -1)$. If $M$ and $m$ denote respectively the maximum and minimum value of $\alpha + \beta$,then the value of $[\frac{M}{m}]$ is- (where $[.]$ denotes the greatest integer function)

$A$ circle of constant radius $a$ passes through the origin $O$ and cuts the coordinate axes at points $P$ and $Q$. The equation of the locus of the foot of the perpendicular from $O$ to $PQ$ is:

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

$A$ circle touches the $x$-axis and also touches the circle with centre at $(0, 3)$ and radius $2$. The locus of the centre of the circle is

The locus of a point $(x, y)$ whose distance from the point $(-g, -f)$ is always $a$,where $k = g^2 + f^2 - a^2$,is: