The equation of the pair of tangents drawn from the origin to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is:

$A$ circle $C_1$ of unit radius lies in the first quadrant and touches both coordinate axes. The radius of another circle that touches both coordinate axes and intersects $C_1$ such that the common chord is of maximum length is:

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is

The length of the common chord of the circles $x^2+y^2-6x+5=0$ and $x^2+y^2+4y-5=0$ is:

Let the tangent to the circle $C_{1}: x^{2}+y^{2}=2$ at the point $M(-1, 1)$ intersect the circle $C_{2}: (x-3)^{2}+(y-2)^{2}=5$ at two distinct points $A$ and $B$. If the tangents to $C_{2}$ at the points $A$ and $B$ intersect at $N$,then the area of the triangle $ANB$ is equal to

Two circles which touch both the coordinate axes intersect at the points $A$ and $B$. If $A=(1,2)$,then $AB=$