Consider the circles ${x^2} + {(y - 1)^2} = 9$ and ${(x - 1)^2} + {y^2} = 25$. They are such that:

Three circles of radii $a, b, c$ $(a < b < c)$ touch each other externally. If they have the $x$-axis as a common tangent,then:

The mid-point of the chord of the circle $x^2+y^2-6x+4y-12=0$ drawn parallel to the tangent at $(-1,1)$ and at a distance of one unit from the tangent is

The line $y = x$ cuts the circle $x^2 + y^2 - 2x = 0$ at points $A$ and $B$. Find the equation of the circle having $AB$ as its diameter.

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

The intercept on the line $y=x$ by the circle $x^2+y^2-2x=0$ is $AB$. The equation of the circle with $AB$ as diameter is: