The number of common tangents of the circles $x^2+y^2-4=0$ and $x^2+y^2-6x-8y-24=0$ is

If a point $P(\alpha, \beta)$ on the line $y=1$ is such that the two distinct chords drawn on $x^2+y^2-\alpha x-y=0$ from $P$ are bisected by the $x$-axis,then

The length of the tangent from the point $(5, 1)$ to the circle $x^2 + y^2 + 6x - 4y - 3 = 0$ is:

In a triangle,two vertices are $(2, 3)$ and $(4, 0)$,and its circumcentre is $(2, z)$ for some real number $z$. The circumradius is

The radius of a circle $C_1$ is thrice the radius of another circle $C_2$. The centres of $C_1$ and $C_2$ are $(1, 2)$ and $(3, -2)$ respectively. If they cut each other orthogonally,find the equation of the circle with radius $r$ and centre $(1, -2)$.

$A$ 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