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 $....\pi$.

Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then $r_1^2+r_2^2-r_1 r_2$ is equal to $...........$

Two circles of radii $4 \text{ cm}$ and $1 \text{ cm}$ touch each other externally and $\theta$ is the angle contained by their direct common tangents. Then $\sin \theta =$

Let $PA$ and $PB$ be the tangent segments drawn from point $P(6, 8)$ to the circle with the centre at origin $O(0, 0)$. The radius $r$ of the circle for which the area of quadrilateral $PAOB$ is maximum, is...

Let $A(2,3)$,$B(3,-1)$,and $C(-3,2)$ be three points. If the centre of the circle passing through $A$,$B$,and $C$ is $O(h, k)$,then find the value of $2k - 4h$.

The area of a circle whose centre is $(h, k)$ and radius $a$ is