The equation of the circle passing through $(2,0)$ and $(0,4)$ and having the minimum radius 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 $\sqrt{2}(\sqrt{3}+1)r > R$

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 a diameter is . . . . . .

If $(x_i, y_i)$ are the vertices of an equilateral triangle $ABC$ such that $(x_1 - 2)^2 + (y_1 - 3)^2 = (x_2 - 2)^2 + (y_2 - 3)^2 = (x_3 - 2)^2 + (y_3 - 3)^2$,then find the value of $2(x_1 + x_2 + x_3) + 3(y_1 + y_2 + y_3)$.

Let $A(2,3)$,$B(3,-1)$,and $C(-3,2)$ be three points. If the centre of the circle passing through $A$,$B$,and $C$ is $O(h, k)$,then find the value of $2k - 4h$.

If the circle $x^2+y^2-6x+2y=28$ cuts off a chord of length $\lambda$ units on the line $2x-5y+18=0$,then the value of $\lambda$ is