The equation of the circle which touches the $x$-axis and whose centre is $(1, 2)$ is

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

If four distinct points $(2k, 3k)$,$(2,0)$,$(0,3)$,and $(0,0)$ lie on a circle,then:

If ${g^2} + {f^2} = c$,then the equation ${x^2} + {y^2} + 2gx + 2fy + c = 0$ will represent

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) -

If the lines $x+2y-5=0$ and $2x-3y+4=0$ lie along diameters of a circle of area $9\pi$,then the equation of the circle is