If in two circles,arcs of the same length subtend angles $30^{\circ}$ and $78^{\circ}$ at the centre,then the ratio of their radii is

The circle$(s)$ touching the $x$-axis at a distance of $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on the $y$-axis is (are):

The point of contact of the circles $x^2+y^2+2x+2y+1=0$ and $x^2+y^2-2x+2y+1=0$ is

The centre of the circle passing through the point $(0, 1)$ and touching the curve $y = x^2$ at $(2, 4)$ is

The equation of the circle which touches the coordinate axes and the line $\frac{x}{3} + \frac{y}{4} = 1$,and whose centre lies in the first quadrant,is ${x^2} + {y^2} - 2cx - 2cy + {c^2} = 0$,where $c$ is:

In a triangle,two vertices are $(2, 3)$ and $(4, 0)$,and its circumcentre is $(2, z)$ for some real number $z$. The circumradius is