Equation of any tangent of the hyperbola with slope $m$ is $y=mx\pm \sqrt{2m^2-4}$
If it passes through $\left(x_1,y_1\right)$ then
$$\begin{gathered} {\left(y_1-m{x}_1\right)}^2=2m^2-4 \\ \Rightarrow \left(x_1^2-2\right)m^2-2x_1{y}_{1}m+\left(y_1^2+4\right)=0 \end{gathered}$$
If $n=m_1,m_2$ then as given $m_1{m}_2=1$

$\Rightarrow \frac{y_1^2+4}{x_1^2-2}=1$
Hence, the required locus is $y^2+4=\left(x^2-2\right)$