Find the equations of the asymptotes for the rectangular hyperbola $2xy + 3x + 4y + 1 = 0$
Equation of given rectangular hyperbola is
$2xy + 3x + 4y + 1 = 0$
i.e. $\;$ $xy + \dfrac{3}{2}x + 2y + \dfrac{1}{2} = 0$
i.e. $\;$ $xy + \dfrac{3}{2}x + 2y + \dfrac{1}{2} + 3 = 3$
i.e. $\;$ $xy + \dfrac{3}{2}x + 2y + \dfrac{7}{2} = 3$
i.e. $\;$ $y \left(x + 2\right) + \dfrac{3}{2} \left(x + 2\right) = 3$
i.e. $\;$ $\left(x + 2\right) \left(y + \dfrac{3}{2}\right) - 3 = 0$
The equation of a rectangular hyperbola differs from the combined equation of asymptotes by a constant.
$\therefore$ $\;$ The combined equation of asymptotes is
$\left(x + 2\right) \left(y + \dfrac{3}{2}\right) = 0$
$\therefore$ $\;$ The separate equations of asymptotes are
$x+ 2 = 0$ $\;$ and $\;$ $y + \dfrac{3}{2} = 0$