Asymptotes of (a+x)²(b²+x²)=x²y²

Curve: (a + x)²(b² + x²) = x²y², i.e. x²y² − (a + x)²(b² + x²) = 0. Expanding the second term gives x⁴ + 2ax³ + (a² + b²)x² + 2ab²x + a²b², so the degree is 4.

Parallel to the y-axis: the coefficient of the highest power of y (y²) is x². Putting x² = 0 gives the asymptote x = 0 (counted twice).

Parallel to the x-axis: the highest power of x is x⁴ with a constant coefficient, so there is none.

Oblique asymptotes:

φ₄(m) = m² − 1
φ₃(m) = −2a

m = ±1. With c = −φ₃/φ₄′ = 2a/(2m) = a/m:

• m = 1: c = a, asymptote y = x + a.
• m = −1: c = −a, asymptote y = −x − a.

(Check: y = ±(a + x)√(b² + x²)/x ≈ ±(x + a)(1 + b²/2x² ...) ≈ ±(x + a).)

Answer: x = 0, y = x + a, y = −(x + a).