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

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3, -3)$ and $(4, -2\sqrt{2})$,and given that $a - 2\sqrt{2}b = 3$,then $(a^{2} + b^{2} + ab)$ is equal to ..... .

If $r_1$ and $r_2$ are the radii of two circles that touch all four circles given by $(x \pm r)^2 + (y \pm r)^2 = r^2$,then find the value of $\frac{r_1+r_2}{r}$.

If the angle between the circles $x^2+y^2-2x-4y+c=0$ and $x^2+y^2-4x-2y+4=0$ is $60^{\circ}$,then $c=$

Let $AB$ be a chord of a circle and $C$ divides $AB$ internally in the ratio $3 : 1$. $A$ line through $C$ cuts the circle at $D$ and $E$ such that the minimum distances of $D$ and $E$ from line $AB$ are $3$ and $2$ respectively. If $r$ is the minimum length of $AB$ such that $\alpha$ is the angle between $AB$ and $DE$ for this $r$,then the value of '$r\alpha$' is

Two concentric circles are given,where the equation of the smaller circle is $x^2 + y^2 = 4$. If each circle makes an intercept on the line $x + y = 2$ and the length of the intercept formed between the two circles is $1$,then the equation of the larger circle is: