If $f\left(x+y\right)=f\left(x\right)·f\left(y\right)$ for all $x$ and $y,f\left(1\right)=2$ and ${\alpha }_n=f\left(n\right),n\in \mathrm{N},$ then equation of the circle having $\left({\alpha }_1,{\alpha }_2\right)$ and $\left({\alpha }_3,{\alpha }_4\right)$ as the ends of its one diameter is

If $\left(m_r,\frac{1}{m_r}\right),r=1,2,3,4$ are concyclic point and $f\left(x\right)=x$, when $x$ is rational and $f\left(x\right)=1-x$, when $x$ is irrational and $a$ is a point where $f\left(x\right)$ is continuous, then $m_1{m}_2{m}_3{m}_4=$