Find the equation of the diameter of the circle $x^2 + y^2 - 4x + 2y - 11 = 0$ which corresponds to the system of parallel chords $x - 2y + c = 0$.

The tangents drawn from a point $(2,-1)$ touch the circle $x^2+y^2+4x-2y+1=0$ at the points $A$ and $B$. If $C$ is the centre of the circle,then the area (in sq. units) of the triangle $ABC$ is

If $P(0,0), Q(1,0)$ and $R\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ are three given points,then the centre of the circle for which the lines $PQ, QR$ and $RP$ are the tangents is

Suppose the tangents drawn to the circle $x^2+y^2-6x-4y-11=0$ from $P(1,8)$ touch the circle at $A$ and $B$. Then the centre of the circle passing through $P, A$ and $B$ is

$A$ line meets the coordinate axes at $A(a, 0)$ and $B(0, b)$. If the perpendicular distances from $A$ and $B$ to the tangent drawn at the origin to the circumcircle of $\triangle OAB$ are $m$ and $n$ respectively,then the diameter of that circle is

The coordinates of the centre of the smallest circle passing through the origin and having $y=x+1$ as a diameter are