$A$ circle is inscribed in an equilateral triangle of side length $6$. Find the area of the square inscribed in this circle.

If $\sin ^{-1}(a)$ is the acute angle between the curves $x^2+y^2=4x$ and $x^2+y^2=8$ at the point $(2,2)$,then $a$ is equal to

The equation of the circle whose radius is $5$ and which touches the circle ${x^2} + {y^2} - 2x - 4y - 20 = 0$ externally at the point $(5, 5)$ is

The number of common tangents to the circles $x^{2}+y^{2}-y=0$ and $x^{2}+y^{2}+y=0$ is

Let $R$ be a rectangle,$C$ be a circle,and $T$ be a triangle in the plane. The maximum possible number of points common to the perimeters of $R, C$ and $T$ is

Let $A(1, 4)$ and $B(1, -5)$ be two points. Let $P$ be a point on the circle $(x-1)^{2} + (y-1)^{2} = 1$ such that $(PA)^{2} + (PB)^{2}$ has a maximum value. Then the points $P, A,$ and $B$ lie on: