Sereinserenormg

2022-01-26

Find the eccentricity of the conic given by:
$\left(x{\mathrm{tan}10}^{\circ }+y{\mathrm{tan}20}^{\circ }+{\mathrm{tan}30}^{\circ }\right)\left(x{\mathrm{tan}120}^{\circ }+y{\mathrm{tan}220}^{\circ }+{\mathrm{tan}320}^{\circ }\right)+2018=0$

ataill0k

Expert

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

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