The locus of the point of intersection of the lines ( 3 )kx + ky -...

The point of intersection of the lines $\left(\sqrt{3}\right)kx + ky - 4\sqrt{3} = 0$ and $\sqrt{3}x - y - 4\left(\sqrt{3}\right)k = 0$.

$k = \dfrac{4\sqrt{3}}{\sqrt{3}x + y} = \dfrac{\sqrt{3}x - y}{4\sqrt{3}}$

$⇒ 3{x}^{2} - {y}^{2} = 48$

$⇒ \dfrac{{x}^{2}}{16} - \dfrac{{y}^{2}}{48} = 1$

Which is a hyperbola.

${b}^{2} = {a}^{2}\left({e}^{2} - 1\right)$

Now, $48 = 16 \left({e}^{2} - 1\right)</p><p>$⇒ e = \sqrt{4} = 2$</p>$