# 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 $n$th degree cannot exceed $n$.