For the equation $ax^2 + 2hxy + 3y^2 + 4x + 8y - 6 = 0$ to represent a circle,what must be the values of $a$ and $h$?

The radius of the circle $x^2 + y^2 + 2x \cos \theta + 2y \sin \theta - 8 = 0$ is

The points $(1,0), (0,1), (0,0)$ and $(2k, 3k), k \neq 0$ are concyclic,if $k$ is

The equation of the circle which has its centre at the point $(3, 4)$ and touches the line $5x + 12y - 11 = 0$ is

Let in an equilateral $\Delta ABC,$ $A(-1 + a \cos \theta, 2 + a \sin \theta),$ $B(-1 + a \cos \alpha, 2 - a \sin \alpha),$ and $C(-1 + a \sin \beta, 2 + a \cos \beta).$ If the length of the median through vertex $A$ is $2b,$ then the equation of the circumcircle of triangle $ABC$ is (where $a$ is a constant) -

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x + y = 1$ is $\sqrt{14}$, then the square of the radius of $C$ is . . . . . .