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 $b=$

Let the position vectors of the vertices of a $\triangle A B C$ be $\mathbf{a , b}, \mathbf{c}$. If on the plane of the triangle, $P$ is a point having position vector $\mathbf{x}$ such that $\mathbf{x} \cdot(\mathbf{c}-\mathbf{b})=\mathbf{a} \cdot \mathbf{c}-\mathbf{a} \cdot \mathbf{b}$ and $\mathbf{x} \cdot(\mathbf{a}-\mathbf{c})=\mathbf{a b}-\mathbf{b} \mathbf{c}$, then for the $\triangle A B C, P$ is the