Is the curve (x^2-a^2)(y^2-b^2)+2xy+3x+4y=7(x2a2)(y2b2)+2xy+3x+4y=7 has only 2 asymptotes parallel to coordinate axes?

1 Answer
May 25, 2017

See below.

Explanation:

Solving for xx (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))x=±(3+2y)+(3+2y)24(y2b2)(a2b2+4ya2y27)2(y2b2)

and solving for yy 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))y=±(4+2x)+(4+2x)24(x2a2)(a2b2+3xb2x27)2(x2a2)

so we have four asymptotes:

y = pm by=±b and

x = pm ax=±a

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

enter image source here