Tangents $OP$ and $OQ$ are drawn from the origin $O$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$. Then,the equation of the circumcircle of the triangle $OPQ$ is

The length of the common chord of the circles $x^2+y^2-6x-4y+9=0$ and $x^2+y^2-8x-6y+23=0$ is

The length of the common chord of the circles $x^2+y^2+3x+5y+4=0$ and $x^2+y^2+5x+3y+4=0$ is

$A$ rhombus is inscribed in the region common to the two circles $x^2+y^2-4x-12=0$ and $x^2+y^2+4x-12=0$. If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area (in square units) of the rhombus is (in $\sqrt{3}$)

Let the tangents drawn from the origin to the circle $x^{2}+y^{2}-8x-4y+16=0$ touch it at the points $A$ and $B$. The $(AB)^{2}$ is equal to

If the circle ${C_1}: {x^2} + {y^2} = 16$ intersects another circle ${C_2}$ of radius $5$ in such a manner that the common chord is of maximum length and has a slope equal to $\frac{3}{4}$,the coordinates of the centre of ${C_2}$ are