Asymptotes of y³+2xy²+x²y−y+1=0

Definition

An asymptote of a curve is a straight line such that the distance of a point P of the curve from the line tends to zero as P moves to infinity along some branch of the curve. (It is a tangent to the curve at infinity.)

Method

For an algebraic curve of degree n, write F = φₙ(x,y) + φₙ₋₁ + φₙ₋₂ + ... and put φₖ(m) = φₖ(1, m).

• Slopes m are the roots of φₙ(m) = 0.
• For a simple root, the intercept c is given by φₙ₋₁(m) + c φₙ′(m) = 0.
• For a repeated root (twice), c is given by c² φₙ″(m)/2! + c φₙ₋₁′(m) + φₙ₋₂(m) = 0.
• Parallel to the x-axis: coefficient of the highest power of x equated to zero.

Solution

F = y³ + 2xy² + x²y − y + 1 = 0, degree 3.

φ₃(m) = m³ + 2m² + m = m(m+1)²
φ₂(m) = 0
φ₁(m) = −m
φ₀    = 1

φ₃(m) = 0 gives m = 0 (simple) and m = −1 (twice).

m = 0: φ₃′(0) = 1, φ₂(0) = 0, so c = 0. Asymptote: y = 0. (Also seen from the coefficient of x², which is y.)

m = −1 (double): φ₃″(m)/2 = 3m + 2 = −1 at m = −1; φ₂′ = 0; φ₁(−1) = 1.

−c² + 0 + 1 = 0, so c = ±1.

Asymptotes: y = −x + 1 and y = −x − 1.

Answer: y = 0, x + y − 1 = 0, x + y + 1 = 0.