Consider the regions $A=\{(x, y) \mid x^2+y^2 \leq 100\}$ and $B=\{(x, y) \mid \sin (x+y)>0\}$ in the plane. Then,the area of the region $A \cap B$ is $....\pi$.

If the length of the tangent from $(f, g)$ to the circle $x^2+y^2=6$ is twice the length of the tangent from the same point to the circle $x^2+y^2+3x+3y=0$,then $f^2+g^2+4f+4g+2$ is equal to

Let $PQ$ be a diameter of the circle $x^{2}+y^{2}=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line $x+y=2$ respectively,then the maximum value of $\alpha \beta$ is

Two parallel chords of a circle of radius $2$ are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the centre,angles of $\frac{\pi}{k}$ and $\frac{2 \pi}{k}$,where $k>0$,then the value of $[k]$ is [Note : $[k]$ denotes the largest integer less than or equal to $k$ ].

The centre of the circle which passes through the vertices of the triangle formed by the lines $y=0$,$y=x$ and $2x+3y=10$ is

Let $AB$ be a line segment of length $2$. Construct a semicircle $S$ with $AB$ as diameter. Let $C$ be the mid-point of the arc $AB$. Construct another semicircle $T$ external to the $\triangle ABC$ with chord $AC$ as diameter. The area of the region inside the semicircle $T$ but outside $S$ is