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:

The length of the common chord of the two circles $(x-a)^2+y^2=a^2$ and $x^2+(y-b)^2=b^2$ is

If $L_1, L_2$ and $L_3$ are the chords of contact of the three points $(2,0), (1,-2)$ and $(4,4)$ respectively with respect to the circle $x^2+y^2=3$,then $L_1, L_2$ and $L_3$ are

If $m$ is the slope and $P(8, \beta)$ is the midpoint of a chord of contact of the circle $x^2+y^2=125$,then the number of values of $\beta$ such that $\beta$ and $m$ are integers is

If $A$ and $B$ are the points of intersection of the circles $x^2+y^2-4x+6y-3=0$ and $x^2+y^2+2x-2y-2=0$,then the distance between $A$ and $B$ is

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius $2$. If the line of the common chord of $C_1$ and $C_2$ intersects the $y$-axis at the point $P$,then the square of the distance of $P$ from the centre of $C_1$ is: