Find the equation of a circle that touches the $Y$-axis at a distance of $4$ units from the origin and cuts an intercept of $6$ units on the $X$-axis.

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

The circle $4x^2+4y^2-12x-12y+9=0$

If the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ touches the $x$-axis,then:

Circles $C_1$ and $C_2$,of radii $r$ and $R$ respectively,touch each other as shown in the figure. The line $l$,which is parallel to the line joining the centres of $C_1$ and $C_2$,is tangent to $C_1$ at $P$ and intersects $C_2$ at $A$ and $B$. If $R^2=2r^2$,then $\angle AOB$ equals

The shortest distance from the line $3x + 4y = 25$ to the circle $x^2 + y^2 - 6x + 8y = 0$ is