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.
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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.
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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.
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The number of asymptotes to an algebraic curve of the $$n$$th degree cannot exceed $$n$$.