$A$ circle touches the $x$-axis and cuts off a chord of length $2l$ from the $y$-axis. The locus of the centre of the circle is

The tangent at any point on the curve $x^2 + y^2 = r^2$ meets the coordinate axes at $A$ and $B$. If lines are drawn through $A$ and $B$ parallel to the coordinate axes to intersect at $P$,find the locus of $P$.

$A$ 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 the line $lx + my = 1$ is a tangent to the circle ${x^2} + {y^2} = {a^2}$,then the locus of the point $(l, m)$ is

The locus of the centre of a circle of radius $2$ which rolls on the outside of the circle ${x^2} + {y^2} + 3x - 6y - 9 = 0$ 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