The number of common tangents to the circles $x^2+y^2-4x-2y+k=0$ and $x^2+y^2-6x-4y+l=0$,having radii $2$ and $3$ respectively,is

The interval of the values of $a$ for which the line $x + y = 0$ bisects $2$ distinct chords drawn from a point $P \left( \frac{1 + \sqrt{2} a}{2}, \frac{1 - \sqrt{2} a}{2} \right)$ to the circle $2x^2 + 2y^2 - (1 + \sqrt{2} a)x - (1 - \sqrt{2} a)y = 0$ is:

The angle subtended by the chord $x+y-1=0$ of the circle $x^2+y^2=1$ at the origin is

Consider the circles $x^2 + (y - 1)^2 = 9$ and $(x - 1)^2 + y^2 = 25$. Which of the following is true?

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

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