The area of the region $\{(x,y) : x^2 + y^2 \leqslant 1 \leqslant x + y\}$ is

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to a larger circle with center $(2,1)$,then the radius of the larger circle is:

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

$A$ circle passing through the point $P(\alpha, \beta)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$. The point $P$ is above the line $AB$. The point $Q$ on the line segment $AB$ is the foot of the perpendicular from $P$ on $AB$. If $PQ$ is equal to $11$ units,then the value of $\alpha \beta$ is $.............$.

The straight line $(x - 2) + (y + 3) = 0$ cuts the circle $(x - 2)^2 + (y - 3)^2 = 11$ at

Let $P(a_1, b_1)$ and $Q(a_2, b_2)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$,then $a_1^2 + a_2^2 + b_1^2 + b_2^2$ is equal to $...........$.