The length of the intercept on the line $4x - 3y - 10 = 0$ by the circle $x^2 + y^2 - 2x + 4y - 20 = 0$ is

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 $\sqrt{2}(\sqrt{3}+1)r > R$

In the figure,$ABCD$ is a unit square. $A$ circle is drawn with centre $O$ on the extended line $CD$ and passing through $A$. If the diagonal $AC$ is tangent to the circle,then the area of the shaded region is

The number of common tangents to the circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-6x-8y-24=0$ is

Let $P$ and $Q$ be the inverse points with respect to the circle $S \equiv x^2+y^2-4x-6y+k=0$ and $C$ be the centre of the circle $S=0$ such that $CP \cdot CQ=4$. If $P=(1,2)$ and $Q=(a, b)$,then $2a=$

Observe the following statements:
$I$. The circle $x^2+y^2-6x-4y-7=0$ touches the $y$-axis.
$II$. The circle $x^2+y^2+6x+4y-7=0$ touches the $x$-axis.
Which of the following is a correct statement?