$A$ circle is inscribed in an equilateral triangle of side length $12$. If the area and perimeter of any square inscribed in this circle are $m$ and $n$,respectively,then $m+n^2$ is equal to

Suppose $Q$ is a point on the circle with centre $P$ and radius $1$,as shown in the figure. $R$ is a point outside the circle such that $QR = 1$ and $\angle QRP = 2^{\circ}$. Let $S$ be the point where the segment $RP$ intersects the given circle. Then,the measure of $\angle RQS$ equals $......^{\circ}$.

If the arcs of the same length in two circles $S_1$ and $S_2$ subtend angles $75^{\circ}$ and $120^{\circ}$ respectively at the center,then the ratio of their radii $\frac{r_1}{r_2}$ is equal to:

The line $4x - 3y + 15 = 0$ intersects the circle $x^2 + y^2 - 6x - 8y = 0$ at two points $A$ and $B$. The maximum area of $\Delta ABC$,where $C$ is a point on the circumference of the circle,will be - .............. $sq. \ units$.

Let a circle $C$ touch the lines $L_{1}: 4x - 3y + K_{1} = 0$ and $L_{2}: 4x - 3y + K_{2} = 0$,where $K_{1}, K_{2} \in R$. If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(-1, 2)$ and $L_{2}$ at $(3, -6)$,then the equation of the circle $C$ is:

The equation of the circle touching the circle $x^2+y^2-6x+6y+17=0$ externally and to which the lines $x^2-3xy-3x+9y=0$ are normal is