If the circles $x^2+y^2-2x+4y+c=0$ and $x^2+y^2+2x-4y+c=0$ have four common tangents,then

The equation of a circle passing through the origin and co-axial to the circles $x^2 + y^2 = a^2$ and $x^2 + y^2 + 2ax = 2a^2$ is

If $m_1$ and $m_2$ are the slopes of the direct common tangents drawn to the circles $x^2+y^2-2x-8y+8=0$ and $x^2+y^2-8x+15=0$,then $m_1+m_2=$

If the straight line $x \cos \alpha + y \sin \alpha = P$ intersects the circle $x^2 + y^2 = a^2$ at $A$ and $B$,then the equation of the circle with diameter $\overline{AB}$ is

The radical centre of the circles $x^2 + y^2 - 16x + 60 = 0$,$x^2 + y^2 - 12x + 27 = 0$,and $x^2 + y^2 - 12y + 8 = 0$ is

Suppose $A(2,3)$ and $B$ are the points of intersection of two circles. The points $P$ lying on one circle and $Q$ lying on the other circle are such that $BP$ and $BQ$ constitute the diameters of the circles. If the slopes of the radical axis and $PQ$ are $3/4$ and $a/b$ respectively,then the value of $3a+4b$ is