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

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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.