If $C_1$ and $C_2$ are the centres of similitude with respect to the circles $x^2+y^2+6x+8y+24=0$ and $x^2+y^2-6x-8y+9=0$,then $C_1C_2=$

Let $C_1$ be the circle of radius $1$ with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$,where $1 < r < 3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Midpoints of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB=\sqrt{5}$,then the value of $r^2$ is

If a tangent to the circle $x^2+y^2+2x+2y+1=0$ is the radical axis of the circles $x^2+y^2+2gx+2fy+c=0$ and $2x^2+2y^2+3x+8y+2c=0$,then

If two circles touch at $(0,0)$ externally,then their radical axis is

Find the radical center of the three circles $x^2 + y^2 = a^2$,$(x - c)^2 + y^2 = a^2$,and $x^2 + (y - b)^2 = a^2$.

If the line $y = x + 3$ intersects the circle $x^2 + y^2 = a^2$ at points $A$ and $B$,then the equation of the circle with $AB$ as its diameter is: