The locus of the midpoints of the chords of the circle $x^2+y^2=16$,which are tangents to the hyperbola $9x^2-16y^2=144$,is

Let $\Gamma$ be a circle with diameter $AB$ and centre $O$. Let $l$ be the tangent to $\Gamma$ at $B$. For each point $M$ on $\Gamma$ different from $A$,consider the tangent $t$ at $M$ and let it intersect $l$ at $P$. Draw a line parallel to $AB$ through $P$ intersecting $OM$ at $Q$. The locus of $Q$ as $M$ varies over $\Gamma$ is

Let a circle of radius $4$ pass through the origin $O$, the points $A(-\sqrt{3}a, 0)$ and $B(0, -\sqrt{2}b)$, where $a$ and $b$ are real parameters and $ab \neq 0$. Then the locus of the centroid of $\Delta OAB$ is a circle of radius

If $A=(1,2), B=(2,1)$ and $P$ is a variable point satisfying the condition $|PA-PB|=3$,then the locus of $P$ is

The locus of the centre of a variable circle which cuts the circles $x^2 + y^2 - 2x - 4y - 1 = 0$ and $x^2 + y^2 - 4x - 2y - 1 = 0$ orthogonally is:

$A$ stick of length $l$ 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