The equation of a circle concentric with the circle $x^2+y^2-6x+12y+15=0$ and having an area that is twice the area of the given circle is

If the circle $x^{2}+y^{2}-2gx+6y-19c=0$,where $g, c \in R$,passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$,then the length of the intercept made by the circle on the $x$-axis is:

The equation of the circle passing through the point $(2, 1)$ and touching the $y$-axis at the origin is:

The number of circles touching the lines $x = 0$,$y = a$ and $y = b$ is

The polar equation of the circle with centre $\left(2, \frac{\pi}{2}\right)$ and radius $3$ units is :

If three distinct points $A, B, C$ are given in the $2$-dimensional coordinate plane such that the ratio of the distance from $(1, 0)$ to the distance from $(-1, 0)$ is equal to $\frac{1}{2}$,then the circumcentre of the triangle $ABC$ is at the point