The angle between the circles $x^2+y^2-2x-9=0$ and $x^2+y^2-4y-1=0$ at their point of intersection is

Let $R$ be the region of the disc $x^2+y^2 \leq 1$ in the first quadrant. Then,the area of the largest possible circle contained in $R$ is

Let $P_1, P_2, P_3, P_4, P_5$ be five equally spaced points on the circumference of a circle of radius $1$,centred at $O$. Let $R$ be the set of points in the plane of the circle that are closer to $O$ than any of $P_1, P_2, P_3, P_4, P_5$. Then,$R$ is a

Let a chord $AB$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $AB$ is $x+y+1=0$,then the equation of the circle $S$ is

If the circles $(x+a)^2+(y+b)^2=a^2$ and $(x+c)^2+(y+d)^2=d^2$ cut orthogonally,then $b(b-2d) =$

Let $A(3,5)$,$B(-2,-7)$ and $C(\alpha, \beta)$ be three points such that $\angle ACB$ is a right angle and the area of triangle $ABC$ is $\frac{82}{3}$ square units. Then the number of such points $C$ is