Find the eccentricity of the conic given by:
$$\left(x\tan 10^\circ+y\tan 20^\circ+\tan 30^\circ\right)\left(x\tan 120^\circ+y\tan 220^\circ+\tan 320^\circ\right)+2018=0$$
What I have tried
$$\bigg(x\tan10^\circ+y\tan 20^\circ+\frac{1}{\sqrt{3}}\bigg)\bigg(\sqrt{3}\; x +y\tan 220^\circ+\tan 320^\circ\bigg)+2018=0$$
$$\begin{align}\Longrightarrow\quad &\sqrt{3}x^2+\sqrt{3}xy\tan 20^\circ+x+xy\tan 10^\circ\tan 220^\circ+y^2\tan 20^\circ\tan 220^\circ \\[4pt] &+\frac{y}{\sqrt{3}}\tan 220^\circ+x\tan 10^\circ\tan 220^\circ+y\tan 20^\circ\tan 320^\circ+\frac{1}{\sqrt{3}}\tan 320^\circ \\[4pt] &+2018=0 \end{align}$$
How do I solve it? Help me, please.