The number of common tangents to the circles $x^2+y^2-x=0$ and $x^2+y^2+x=0$ is/are:

$A$ line is drawn through a fixed point $P(\alpha, \beta)$ to cut the circle $x^{2}+y^{2}=r^{2}$ at $A$ and $B$. Then $PA \cdot PB$ is equal to

If $\theta$ is the angle between the tangents from $(-1, 0)$ to the circle $x^2+y^2-5x+4y-2=0$,then $\theta$ is equal to

Let $PQ$ be a diameter of the circle $x^{2}+y^{2}=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line $x+y=2$ respectively,then the maximum value of $\alpha \beta$ is

From a point $P$ on the line $4x - 3y = 6$,two tangents are drawn to the circle $x^2 + y^2 - 6x - 4y + 4 = 0$. If the angle between these tangents is $\tan^{-1}\left(\frac{24}{7}\right)$,then $P$ can be:

The point of contact of the circles $x^2+y^2+2x+2y+1=0$ and $x^2+y^2-2x+2y+1=0$ is