Question

Is the curve `(x^2-a^2)(y^2-b^2)+2xy+3x+4y=7` has only 2 asymptotes parallel to coordinate axes?

Answer 1

See below.

Explanation:

Solving for `x` (using the Bashkara formula) we have

#x = (pm(3 +2 y) + sqrt[(3 + 2 y)^2 - 4 (y^2-b^2) (a^2 b^2 + 4 y - a^2 y^2-7)])/( 2 (y^2-b^2))#

and solving for `y` we have

#y=(pm(4 + 2 x) + sqrt[(4 + 2 x)^2 - 4 (x^2-a^2) (a^2 b^2 + 3 x - b^2 x^2-7)])/(2 (x^2-a^2))#

so we have four asymptotes:

`y = pm b` and

`x = pm a`

Attached a plot for `a=1`, `b=2`