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

If $(\alpha, \beta)$ is the centre of a circle passing through the origin,then its equation is

The lengths of the intercepts made by a circle $S$ on $X$ and $Y$-axes are $\frac{2 \sqrt{13}}{3}$ and $\frac{2 \sqrt{22}}{3}$ respectively. If the radius of the circle $S$ is $\frac{\sqrt{38}}{3}$ and its centre $C$ lies in the second quadrant,then $C=$

Find the equation of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and having its center at $(0, 3)$.

If a circle passes through points $(4,0)$ and $(0,2)$ and its centre lies on the $Y$-axis. If the radius of the circle is $r$,then the value of $r^2-r+1$ is