Let the tangents at two points $A$ and $B$ on the circle $x^{2} + y^{2} - 4x + 3 = 0$ meet at the origin $O(0,0)$. Then the area of the triangle $OAB$ 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 $\sqrt{3}$)

The equation of the common tangent to the circles $x^2+y^2-4x+10y+20=0$ and $x^2+y^2+8x-6y-24=0$ is

If $P$ and $Q$ are the points of intersection of the circles $x^2 + y^2 + 3x + 7y + 2p - 5 = 0$ and $x^2 + y^2 + 2x + 2y - p^2 = 0$,then there is a circle passing through $P, Q$ and $(1, 1)$ for:

If $\frac{2}{\sqrt{5}}$ is the length of the common chord of the circles $x^2+y^2+2x+2y+1=0$ and $x^2+y^2+\alpha x+3y+2=0, \alpha \neq 0$,then $\alpha=$

The distance between the chords of contact of the tangents to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ from the origin and the point $(g, f)$ is