The locus if midpoint of focal chords of the hyperbola  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose eccentricity is $e_1$  lies on another hyperbola whose eccentricity is e₂ then$\frac{e_1}{e_2}$   is

P  $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ is midpoint of chord of hyperbola, then equations of chord is S₁=S₁₁ 
$\Rightarrow \frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}-1=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$
focus (a$e_1$,0)  lies on this equation
$$\begin{gathered} \Rightarrow \frac{{\mathrm{e}}_1{\mathrm{x}}_1}{\mathrm{a}}=\frac{{{\mathrm{x}}_1}^2}{{\mathrm{a}}^2}-\frac{{{\mathrm{y}}_1}^2}{{\mathrm{b}}^2} \\ Locus is \Rightarrow \frac{{\left(\mathrm{x}-\frac{{\mathrm{ae}}_1}{2}\right)}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\frac{{\mathrm{e}}_1^2}{4} \end{gathered}$$
Then ${\mathrm{e}}_2={\mathrm{e}}_1$