If the area of an equilateral triangle inscribed in the circle $x^2 + y^2 + 10x + 12y + c = 0$ is $27\sqrt{3} \text{ sq. units}$,then $c$ is equal to

Given two circles $x^2+y^2+8x-6y-24=0$ and $x^2+y^2-4x+10y+20=0$. Then they are

The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then,the angle at the centre of the sector in radians is

If the line $x + 2by + 7 = 0$ is a diameter of the circle $x^2 + y^2 - 6x + 2y = 0$,then $b = $

For the two circles $x^2+y^2=16$ and $x^2+y^2-2y=0$, there is/are:

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