If from the origin two tangents are drawn to the circle $(x - 2)^2 + y^2 = 1$,then the length of the chord of contact is-

The distance from the centre of the circle $x^2 + y^2 = 2x$ to the straight line passing through the points of intersection of the two circles $x^2 + y^2 + 5x - 8y + 1 = 0$ and $x^2 + y^2 - 3x + 7y - 25 = 0$ is-

Let $P$ be any point on the circle $x^2+y^2-2x-1=0$ and $C$ be its centre. Let $AB$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2x=0$. Then the locus of the circumcentre of the triangle $CAB$ 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

The common chord of the circles $x^2 + y^2 + 4x + 1 = 0$ and $x^2 + y^2 + 6x + 2y + 3 = 0$ is

Given that $a > 2b > 0$ and that the line $y = mx - b \sqrt{1 + m^2}$ is a common tangent to the circles $x^2 + y^2 = b^2$ and $(x - a)^2 + y^2 = b^2$. Then the positive value of $m$ is