Equations of Asymptotes

The results and formulas related to asymptotes are listed below.

Asymptotes
a.  Obtain {\phi _n}\left( m \right) by putting x = 1, y = m in the highest degree terms of the equation of the curve.
b.  Obtain {\phi _{n - 1}}\left( m \right) by putting x = 1, y = m in the {\left( {n - 1} \right)^{{\text{th}}}} degree terms of the equation of the curve.
c. Obtain {\phi _{n - 2}}\left( m \right) by putting x = 1, y = m in the {\left( {n - 2} \right)^{{\text{th}}}} degree terms of the equation of the curve.
d. Put {\phi _n}\left( m \right) = 0 and solve it for m. Let {m_1},{m_2},{m_3} etc., be its roots, then:
e. C = - \frac{{{\phi _{n - 1}}\left( m \right)}}{{{{\phi '}_n}\left( m \right)}} where {\phi '_n}\left( m \right) \ne 0.
f.  If {\phi '_n}\left( m \right) = 0, then c is obtained by \frac{{{c^2}}}{{2!}}{\phi ''_n}\left( m \right) + c{\phi '_{n - 1}}\left( m \right) + {\phi _{n - 2}}\left( m \right) = 0 asymptotes are then y = mx + c asymptotes parallel to x-axis.

  1. The asymptotes parallel to the x-axis are obtained by equating to zero, the coefficients of the highest power of x in the equation of curve.
  2. The asymptotes parallel to the y-axis are obtained by equating to zero, the coefficients of the highest power of y in the equation of curve.
  3. The number of asymptotes to an algebraic curve of the nth degree cannot exceed n.