If the ratio of the lengths of tangents drawn from the point $(f, g)$ to the circles $x^2 + y^2 = 6$ and $x^2 + y^2 + 3x + 3y = 0$ is $2 : 1$,then:

The number of real circles cutting orthogonally the circle $x^{2}+y^{2}+2x-2y+7=0$ is

If the point of intersection of the pair of the transverse common tangents and that of the pair of direct common tangents drawn to the circles $x^2+y^2-14x+6y+33=0$ and $x^2+y^2+30x-2y+1=0$ are $T$ and $D$ respectively,then the centre of the circle having $TD$ as diameter is

$A$ line drawn through the point $P(4, 7)$ cuts the circle $x^2 + y^2 = 9$ at the points $A$ and $B$. Then $PA \cdot PB$ is equal to

The values of the constant term in the equation of a circle passing through $(1, 2)$ and $(3, 4)$ and touching the line $3x + y - 3 = 0$ are

If the length of the tangents drawn from the point $(1, 2)$ to the circles $x^2 + y^2 + x + y - 4 = 0$ and $3x^2 + 3y^2 - x - y + k = 0$ are in the ratio $4 : 3$,then $k =$