In a $\triangle A B C$, if $\sin \frac{A}{2}=\frac{1}{4} \sqrt{\frac{3}{5}}, a=2, c=5$ and $b$ is an integer, then the area (in sq. units) of $\triangle A B C$ is

In a $\triangle A B C$ if $a+c=5 b$, then $\cot \frac{A}{2} \cot \frac{C}{2}=$

In a $\triangle A B C$, if $r_1=3, r_2=4, r_3=6$, then $b=$

In $\triangle A B C$, the sum of the lengths of two sides is $x$ and the product of those lengths is $y$. If $c$ is the length of its third side and $x^2-c^2=y$, then the circumradius of that triangle is