$f(\displaystyle \mathrm{m}_{\mathrm{i}}, \frac{1}{\mathrm{m}_{\mathrm{i}}})$ , $\mathrm{i}=1,2,3,4$ are four distinct points on the circle with centre origin, then value of $\mathrm{m}_{1}\mathrm{m}_{2}\mathrm{m}_{3}\mathrm{m}_{4}$ is equal to

Equation of circle having centre of origin $(0,0)$ and radius $=r$,

$S_{1};x^{2}+y^{2}=r^{2}----(1)$

$\therefore f(m_{1},\dfrac{1}{m_{2}}),f(m_{2},\dfrac{1}{m_{2}}),--f(m_{4},\dfrac{1}{m_{4}})$

These points lie on $S_{1}$.

Let $\displaystyle f(m, \frac{1}{m})$ is point lie on S_{1},

$m^{2}+\dfrac{1}{m^{2}}=r^{2}$

$m^{4}+1-r^{2}m^{2}=0$

$m^{4}-1-r^{2}m^{2}+1=0$

$m_{1},m_{2},m_{3}$ and $m_{4}$ are roots of this equation

So, $m_{1}m_{2}m_{3}m_{4} =1$