Find the asymptotes of the hyperbola \(2x^2 + 5xy + 2y^2 + 4x + 5y\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.

Solution :

Let \(2x^2 + 5xy + 2y^2 + 4x + 5y + k\) = 0 be asymptotes. This will represent two straight line

so \(abc + 2fgh – af^2 – bg^2 – ch^2\) = 0 \(\implies\) 4k + 25 – \(25\over 2\) – 8 – \(25\over 4\)k = 0

\(\implies\) k = 2

\(\implies\) \(2x^2 + 5xy + 2y^2 + 4x + 5y + 2\) = 0 are asymptotes

\(\implies\) (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes

and \(2x^2 + 5xy + 2y^2 + 4x + 5y + c\) = 0 is general equation of hyperbola.