Algebra Chapter XI: Linear Asymptotes

Definition of Asymptotes

• A straight line that intersects a curve at two points infinitely far from the origin, without being entirely at infinity, is classified as an asymptote.

Finding Equations of Asymptotes

• The equation of a plane algebraic curve of nth degree can be expressed in a homogeneous form:
  ( x^n \phi_n \left(\frac{Y}{X}\right) + x^{n-1} \phi_{n-1} \left(\frac{Y}{X}\right) + ... = 0 ).

• To determine where a straight line ( y = mx + c ) intersects this curve, substitute ( \frac{Y}{X} = m + \frac{c}{X} ) into the curve's equation.

• This leads to an equation:
  ( x^n \phi_n \left(m + \frac{c}{X}\right) + x^{n-1} \phi_{n-1} \left(m + \frac{c}{X}\right) + ... = 0 ).

Taylor's Expansion

• By employing Taylor's theorem, expand the terms of the new equation, resulting in:
  ( x^n \phi_n (m) + x^{n-1} [c \phi_n'(m) + \phi_{n-1} (m)] + ... = 0 ).

• This equation is of degree n in x; thus, the line can intersect the curve at n points that may be either real or imaginary.

Conditions for Asymptotes

• For ( y = mx + c ) to be an asymptote, the following conditions must be satisfied:
  • ( \phi_n (m) = 0 ) (i.e., the coefficient of the highest power of x is zero), indicating the line cuts the curve at infinity.
  • ( c \phi_n'(m) + \phi_{n-1}(m) = 0 ) (i.e., setting the coefficient of ( x^{n-1} ) to zero).

Roots and Corresponding Values

• The equation for obtaining values of m yields n roots, denoted as:
  ( m_1, m_2, ... , m_n ).

• The corresponding values of c for each root are calculated as:
  ( c_1 = -\frac{\phi_{n-1}(m_1)}{\phi_n'(m_1)}, c_2 = -\frac{\phi_{n-1}(m_2)}{\phi_n'(m_2)}, ... ).

Resulting Asymptotes

• The resulting equations of the asymptotes take the form:
  • ( y = m_1 x - \frac{\phi_{n-1}(m_1)}{\phi_n'(m_1)} )
  • Further asymptotes follow similarly by substituting corresponding ( m_k ) values.