Asymptotes of x(x−y)²−3(x²−y²)+8y=0

Curve: x(x − y)² − 3(x² − y²) + 8y = 0. Expanding gives x³ − 2x²y + xy² − 3x² + 3y² + 8y = 0, degree 3.

φ₃(m) = (1 − m)²
φ₂(m) = −3(1 − m²) = 3m² − 3
φ₁(m) = 8m

Parallel to y-axis: the coefficient of the highest power of y (y²) is x + 3. Putting it to zero gives the asymptote x = −3. (φ₃ has degree 2 in m, which agrees with one asymptote parallel to the y-axis.)

Slopes: φ₃(m) = (1 − m)² = 0 gives m = 1 twice.

For the double root: φ₃″(m)/2 = 1, φ₂′(1) = 6m = 6, φ₁(1) = 8.

c² + 6c + 8 = 0, so (c + 2)(c + 4) = 0, c = −2 or −4.

Asymptotes: y = x − 2 and y = x − 4.

Answer: x + 3 = 0, y = x − 2, y = x − 4.