If the radical axis of the circles $x^2+y^2+2 \alpha x+2 \beta y+c=0$ and $x^2+y^2+\frac{3}{2} x+4 y+c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$,then $4 \alpha \beta-8 \alpha-3 \beta+10=$

The circles $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by+c=0$ touch each other externally,if

The equation of the tangent at the point $(0, 3)$ on the circle which cuts the circles $x^2 + y^2 - 2 x + 6 y = 0$,$x^2 + y^2 - 4 x - 2 y + 6 = 0$ and $x^2 + y^2 - 12 x + 2 y + 3 = 0$ orthogonally is

$A$ circle $S$ cuts three circles $x^2+y^2-4x-2y+4=0$,$x^2+y^2-2x-4y+1=0$,and $x^2+y^2+4x+2y+1=0$ orthogonally. Then,the radius of $S$ is

$A$ circle $S$ passes through the points of intersection of the circles $x^2+y^2-2x-3=0$ and $x^2+y^2-2y=0$. If $x+y+1=0$ is a tangent to the circle $S$,then the equation of $S$ is

Let $A(1, 2)$ be the centre and $3$ be the radius of a circle $S$. Let $B(-1, -1)$ be the centre and $r$ be the radius of another circle $S^{\prime}$. If $\frac{\pi}{3}$ is the angle between the circles $S$ and $S^{\prime}$,then the number of possible values of $r$ is