Find the eccentricity of the conic given by: (xtan⁡10∘+ytan⁡20∘+tan⁡30∘)(xtan⁡120∘+ytan⁡220∘+tan⁡320∘)+2018=0

Equating to 0 the expressions inside the parentheses we get the equations of two lines, which are the asymptotes of the hyperbola:
${\displaystyle x{\tan 10}^{\circ }+y{\tan 20}^{\circ }+{\tan 30}^{\circ }=0,-x{\tan 60}^{\circ }+y{\tan 40}^{\circ }+{\tan 320}^{\circ }=0}$,
where I used ${\displaystyle {\tan 120}^{\circ }=-{\tan 60}^{\circ }}$ and ${\displaystyle {\tan 220}^{\circ }={\tan 40}^{\circ }}$
But these lines are perpendicular, because
${\displaystyle {\tan 10}^{\circ }{\tan 60}^{\circ }={\tan 20}^{\circ }{\tan 40}^{\circ }}$
(remembering that ${\displaystyle {\tan 60}^{\circ }=\frac{1}{{\tan 30}^{\circ }}}$).
Hence this is a rectangular hyperbola and its eccentricity is ${\displaystyle \sqrt{2}}$.