Let mid point be $(\mathrm{h}, \mathrm{k})$

chord of hyperbola: $\mathrm{hx}-\mathrm{ky}=\mathrm{h}^{2}-\mathrm{k}^{2}$

this is tangent to $\mathrm{x}^{2}=4 \mathrm{by}$

form of tangent for parabola: $\mathrm{y}=\mathrm{mx}-\mathrm{bm}^{2}$

comparing we get

$\mathrm{m}=\frac{\mathrm{h}}{\mathrm{k}} ;-\mathrm{bm}^{2}=-\left(\frac{\mathrm{h}^{2}-\mathrm{k}^{2}}{\mathrm{k}}\right)$

$\therefore \mathrm{b}\left(\frac{\mathrm{h}^{2}}{\mathrm{k}^{2}}\right)=-\left(\frac{\mathrm{h}^{2}-\mathrm{k}^{2}}{\mathrm{k}}\right)$

clearly, locus is dependent on $b,$ but not $a$