Among all cyclic quadrilaterals inscribed in a circle of radius $R$ with one of its angles equal to $120^{\circ}$,consider the one with the maximum possible area. Its area is

If the ratio of the lengths of the tangents drawn from the point $(1, 2)$ to the circles $x^2 + y^2 + x + y - 4 = 0$ and $3x^2 + 3y^2 - x - y + k = 0$ is $4 : 3$,then $k = \dots$

The number of common tangents of the circles given by $x^2 + y^2 - 8x - 2y + 1 = 0$ and $x^2 + y^2 + 6x + 8y = 0$ is

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8x-6y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$,then the length of the line segment $AC$ is.

Find the number of common tangents that can be drawn to the circles $x^2 + y^2 - 4x - 6y - 3 = 0$ and $x^2 + y^2 + 2x + 2y + 1 = 0$.

Suppose a circle passes through $(0, a)$ and $(b, h)$ having its centre at $(c, 0)$. Then the value of $c$ is