The number of circles touching the line $y - x = 0$ and the $y$-axis 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

The locus of the points on the curve ${y^2 = 4a(x + a \sin \frac{x}{a})}$ where the tangent is parallel to the $x$-axis represents . . . .

The two points $A$ and $B$ in a plane are such that for all points $P$ lying on a circle satisfying $\frac{PA}{PB} = k$,then $k$ will not be equal to:

The locus of a point which moves in such a way that its distance from $(0,0)$ is three times its distance from the $x$-axis is given by:

The point of concurrence of all the chords of the curve $3x^2 - y^2 - 2x + 4y = 0$ which subtend a right angle at the origin is