$If \left(m_i,\frac{1}{m_i}\right), i=1,2,3,4 are concyclic points, then the value of m_1{m}_2{m}_3{m}_4 is$

$Letequationofcircle{x}^2+y^2+2gx+2fy+c=0$

$\left(m,\frac{1}{m}\right) lies on circle$

$\therefore m^2+\frac{1}{m^2}+2gm+2f\left(\frac{1}{m}\right)+c=0$

$\Rightarrow m^4+2g{m}^3+c{m}^2+2fm+1=0\to \left(1\right)$

$Let m_1, m_2, m_3, m_4 are roots of equation\left(1\right)$

$Product of roots m_1\cdot m_2\cdot m_3\cdot m_4=1$