The locus of the point of intersection

$\sqrt{3} \mathrm{kx}+\mathrm{ky}=4 \sqrt{3}$

$\sqrt{3} \mathrm{kx}-\mathrm{ky}=4 \sqrt{3} \mathrm{k}^{2}$

Adding equation (1) $\backslash \&(2) 2 \sqrt{3} \mathrm{kx}=4 \sqrt{3}\left(\mathrm{k}^{2}+1\right)$

$x=2\left(k+\frac{1}{k}\right)$

Substracting equation (1) $\backslash \&(2) y=2 \sqrt{3}\left(\frac{1}{k}-k\right)$

$\therefore \frac{\mathrm{x}^{2}}{4}-\frac{\mathrm{y}^{2}}{12}=4$

$\frac{x^{2}}{16}-\frac{y^{2}}{48}=1 \quad$ Hyperbola

$\therefore \mathrm{e}^{2}=1+\frac{48}{16}$

$e=2$