The length of the diameter of a circle which touches the $x$-axis at the point $(1, 0)$ and passes through the point $(2, -3)$ is:

Suppose $Q$ is a point on the circle with centre $P$ and radius $1$,as shown in the figure. $R$ is a point outside the circle such that $QR = 1$ and $\angle QRP = 2^{\circ}$. Let $S$ be the point where the segment $RP$ intersects the given circle. Then,the measure of $\angle RQS$ equals $......^{\circ}$.

The absolute difference between the squares of the radii of the two circles passing through the point $(-9, 4)$ and touching the lines $x+y=3$ and $x-y=3$ is equal to . . . . . . .

If $Q(h, k)$ is the inverse point of the point $P(1, 2)$ with respect to the circle $x^2+y^2-4x+1=0$,then $2h+k=$

The length of the transverse common tangent of the circles $x^2+y^2-2x+4y+4=0$ and $x^2+y^2+4x-2y+1=0$ is

Let a chord $AB$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $AB$ is $x+y+1=0$,then the equation of the circle $S$ is