$A$ 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

$A(4,3)$ and $B(2,5)$ are two points. If $P$ is a variable point on the same side as the origin with respect to the line $AB$ and is at most at a distance of $5$ units from the midpoint of $AB$,then the locus of $P$ is

The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ meet the positive coordinate axes at points $A$ and $B$. Then the minimum value of $OA+OB$,where $O$ is the origin,is equal to

Any circle passing through the point of intersection of the lines $x + \sqrt{3}y = 1$ and $\sqrt{3}x - y = 2$ intersects these lines at points $P$ and $Q$. The angle subtended by the arc $PQ$ at its centre is- ............. $^o$

The shortest distance between the curves $y^2 = x^3$ and $9x^2 + 9y^2 - 30y + 16 = 0$ is