The circles $x^2 + y^2 - 10x + 16 = 0$ and $x^2 + y^2 = r^2$ intersect each other in two distinct points,if

Let $G$ be a circle of radius $R>0$. Let $G_1, G_2, \ldots, G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1, G_2, \ldots, G_n$ touches the circle $G$ externally. Also,for $i=1,2, \ldots, n-1$,the circle $G_i$ touches $G_{i+1}$ externally,and $G_n$ touches $G_1$ externally. Then,which of the following statements is/are $TRUE$?
$(A)$ If $n=4$,then $(\sqrt{2}-1)r < R$
$(B)$ If $n=5$,then $r < R$
$(C)$ If $n=8$,then $(\sqrt{2}-1)r < R$
$(D)$ If $n=12$,then $\sqrt{2}(\sqrt{3}+1)r > R$

If $P(0,0), Q(1,0)$ and $R\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ are three given points, then the centre of the circle for which the lines $PQ, QR$ and $RP$ are the tangents is

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2 + y^2 - 4x - 4y + 6 = 0$ and $x^2 + y^2 - 10x - 10y + \lambda = 0$ is the interval:

If a circle $S$ passing through the points $A(1, 2)$ and $B(2, 1)$ has its centre $C$ located in the third quadrant at a distance of $\frac{7}{\sqrt{2}}$ units from the line $AB$,then the point $P(1, -2)$:

If $(3,-2)$ is the centre of the circle $S \equiv x^2+y^2+2gx+2fy-23=0$ and $A$ is a point on the circle $S=0$ such that its distance from a point $P(-1,-5)$ is least,then $A=$