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 equation of the circle passing through $(2,0)$ and $(0,4)$ and having the minimum radius is:

Let $C_1$ and $C_2$ be two circles touching each other externally at point $A$. Let $AB$ be the diameter of circle $C_1$. Draw a secant $BA_3$ to circle $C_2$,intersecting circle $C_1$ at point $A_1$ (where $A_1 \neq A$) and circle $C_2$ at points $A_2$ and $A_3$. If $BA_1 = 2$,$BA_2 = 3$,and $BA_3 = 4$,then the radii of circles $C_1$ and $C_2$ are respectively:

Suppose that two chords,drawn from the point $(1, 2)$ on the circle $x^2 + y^2 + x - 3y = 0$,are bisected by the $y$-axis. If the other ends of these chords are $R$ and $S$,and the midpoint of the line segment $RS$ is $(\alpha, \beta)$,then $6(\alpha + \beta)$ is equal to:

If the circle $x^2 + y^2 = a^2$ cuts a chord of length $2b$ on the line $y = mx + c$,then:

The number of common tangents to the circles $x^2+y^2-18x-15y+131=0$ and $x^2+y^2-6x-6y-7=0$ is: