Question

Find the asymptotes of the curve
x² y²-4 (x + y ² ) -8 (x+y) + 16=0
which are parallel to x-axis and y-axis​

Answer 1

Answer:

y = -2 and y = +2.

Step-by-step explanation:

Equation of the curve is y⁴ + x² y² + 2 x y² - 4 x² - y + 1 = 0.

=> (y²-4) x² + 2y² x + (y⁴ - y + 1) = 0.

=> x = [ -y² +- √{y⁴ - (y²-4)(y⁴ -y+1)} ] / [y² - 4).

=> x =  -y² /(y² - 4) + - √(5y⁴- y⁶+y³-y²-4y+4) / (y² - 4)

The asymptotes obtained from the two factors in the denominator are y = 2 and y = -2.

The term under the root becomes negative for large values of y. So there are no asymptotes for y = infinity or - infinity.