The figure shows $\Delta ABC$ with $AB = 3, AC = 4$ and $BC = 5$. Three circles $S_1, S_2$ and $S_3$ have their centers at $A, B$ and $C$ respectively and they touch each other externally. The sum of the areas of the three circles is: (in $\pi$)

Let $n \geq 3$ and let $C_1, C_2, \ldots, C_n$ be circles with radii $r_1, r_2, \ldots, r_n$,respectively. Assume that $C_i$ and $C_{i+1}$ touch externally for $1 \leq i \leq n-1$. It is also given that the $X$-axis and the line $y=2 \sqrt{2} x+10$ are tangential to each of the circles. Then,$r_1, r_2, \ldots, r_n$ are in

Let $S$ be a subset of the plane defined by $S = \{(x, y) : |x| + 2|y| = 1\}$. Then,the radius of the smallest circle with centre at the origin and having non-empty intersection with $S$ is

The number of common tangents to the circles $x^2 + y^2 - 2x - 1 = 0$ and $x^2 + y^2 - 2y - 7 = 0$ is:

The angle between two circles,each passing through the centre of the other,is

Let $A(1, 4)$ and $B(1, -5)$ be two points. Let $P$ be a point on the circle $(x-1)^{2} + (y-1)^{2} = 1$ such that $(PA)^{2} + (PB)^{2}$ has a maximum value. Then the points $P, A,$ and $B$ lie on: