Find the area of a circle that passes through the point $(4, 6)$ and has its center at $(1, 2)$. (in $\pi$)

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 acute angle between the curves $x^2+y^2=x+y$ and $x^2+y^2=2y$ is

If the number of common tangents to the pair of circles $x^2+y^2-2x+4y-4=0$ and $x^2+y^2+4x-4y+\alpha=0$ is $4$,then the least integral value of $\alpha$ is

Let $XY$ be the diameter of a semi-circle with center $O$. Let $A$ be a variable point on the semi-circle and $B$ another point on the semi-circle such that $AB$ is parallel to $XY$. The value of $\angle BOY$ for which the inradius of $\triangle AOB$ is maximum,is

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the midpoint of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$,then the radius of the circle $C$ is