$A$ circle $S$ touches the $Y$-axis at $(0,3)$ and makes an intercept of length $8$ units on the $X$-axis. If the centre $C$ of the circle $S$ lies in the second quadrant,then the distance of $C$ from the point $(-2,-1)$ is

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

Given $r_1, r_2 > 0$ and $C_1, C_2$ are the centres of two circles having only two common tangents. If $C_1 C_2 = r_1 + r_2$,which of the following is correct?

If a circle $C$ passing through the point $(4, 0)$ touches the circle $x^2+y^2+4x-6y=12$ externally at the point $(1, -1)$,then the radius of $C$ is

The minute hand of a watch is $1.5 \, \text{cm}$ long. How far does its tip move in $40$ minutes (in $, \text{cm}$)? (Use $\pi = 3.14$).

$A$ straight line meets the coordinate axes at $A$ and $B$. $A$ circle is circumscribed about the triangle $OAB$,where $O$ is the origin. If $m$ and $n$ are the distances of the tangent to the circle at the origin from the points $A$ and $B$ respectively,then the diameter of the circle is