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

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

The length of the chord intercepted by the circle $x^2+y^2-4x+6y-12=0$ on the line $4x+3y+1=0$ is

Find the equation of the circle which touches the $X$-axis at a distance of $+3$ from the origin and cuts an intercept of $8$ on the positive $Y$-axis.

$A$ semi-circle of diameter $1$ unit sits at the top of a semi-circle of diameter $2$ units. The shaded region inside the smaller semi-circle but outside the larger semi-circle is called a lune. The area of the lune is

Let $C$ be the circle $x^2+y^2=1$ in the $XY$-plane. For each $t \geq 0$,let $L_t$ be the line passing through $(0,1)$ and $(t, 0)$. Note that $L_t$ intersects $C$ in two points,one of which is $(0,1)$. Let $Q_t$ be the other point. As $t$ varies between $1$ and $1+\sqrt{2}$,the collection of points $Q_t$ sweeps out an arc on $C$. The angle subtended by this arc at $(0,0)$ is