The length of the chord intercepted by the circle $x^2+y^2+2x+4y-20=0$ on the line $3x+4y-6=0$ is

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to a larger circle with center $(2,1)$,then the radius of the larger circle 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

The line $4x - 3y + 2 = 0$ intersects the circle $x^2 + y^2 - 2x + 6y + c = 0$ at two points $A$ and $B$,and the length of the chord $AB = 8$. If $(1, k)$ is a point on the given circle and $k > 0$,then $k =$

Let $\theta$ be the angle between the circles $S \equiv x^2+y^2+2x-2y+c=0$ and $S' \equiv x^2+y^2-6x-8y+9=0$. If $c$ is an integer and $\cos \theta = \frac{5}{16}$,then the radius of the circle $S=0$ is

Let $A(2, 3), B(4, 5)$ be two points and let $C = (x, y)$ be a point such that $(x - 2)(x - 4) + (y - 3)(y - 5) = 0$. If the area of $\Delta ABC = \sqrt{2}$ sq. unit,then the maximum number of positions of $C$ in the $xy$ plane is