Question

The asymptotes of the hyperbola $ xy - 3x + 4y + 2 = 0 $ are:
A. $ x = - 4 $
B. $ x = 4 $
C. $ y = - 3 $
D. $ y = 3 $

Answer 1

Hint: Take the general equation of the asymptote of the hyperbola and then compare the coefficients of the $ x $ term and $ y $ term in the equation of hyperbola for the asymptotes parallel to $ x $ axis and $ y $ axis.

Complete step-by-step answer:
The equation of the hyperbola is given by
 $ xy - 3x + 4y + 2 = 0 $ .
The equation of the hyperbola given in the question is
 $ xy - 3x + 4y + 2 = 0 $ .
For the asymptote of the hyperbola parallel to the $ x $ axis , take the coefficient of the highest degree of $ x $ in the equation to zero.
The equation of the hyperbola written in another form is
 $ \left( {y - 3} \right)x + 4y + 2 = 0 $ .
The coefficient of the highest degree of $ x $ in the equation is equal to $ y - 3 $ .
So, the equation of the asymptote parallel to $ x $ -axis for the hyperbola
 $ xy - 3x + 4y + 2 = 0 $ is given by $ y - 3 = 0 $ or it can also be written as $ y = 3 $ .
For the asymptote of the hyperbola parallel to the $ y $ axis , take the coefficient of the highest degree of $ y $ in the equation to zero.
The equation of the hyperbola written in another form is
 $ \left( {x + 4} \right)y - 3x + 2 = 0 $ . The coefficient of the highest degree of $ y $ in the equation is equal to $ x + 4 $ .
So, the equation of the asymptote parallel to $ y $ -axis for the hyperbola
 $ xy - 3x + 4y + 2 = 0 $ is given by $ x + 4 = 0 $ or it can also be written as $ x = - 4 $ .
So, the two asymptotes of the hyperbola $ xy - 3x + 4y + 2 = 0 $ are $ x = - 4 $ and $ y = 3 $ .
So, the correct answer is “Option A and ”D.

Note: The equation of the asymptote parallel to the coordinate axis is given by equating the coefficient of the respective $ x $ and $ y $ variable of the highest degree in the equation of the hyperbola to zero.