## Asmptotes of Hyperbola From Algebraic Fraction Form

Given a hyperbola written as an algebraic fraction,\[y=\frac{ax+b}{cx+d}\]

, we can find the equations of the asymptotes using long division and rearranging into the form \[(x-x_0)(y-y_0=K\]

. The asymptotes are then \[x=x_0, \; y=y_0\]

. If \[K \gt 0\]

the hyperbola is in the 1st and 3rd quadrant and if \[K \lt 0\]

the hyperbola is in the 2nd and 4th quadrant. Example: \[y=\frac{3x-2}{x+1}\]

.Long division gives

\[y=3- \frac{5}{x+1}\]

.Subtract 3 to give

\[y-3=- \frac{5}{x+1}\]

.Multiply by

\[(x+1)\]

to give \[(x+1)(y-3)=-5\]

.The asymptotes are

\[x=-1, \; y=3\]

and the hyperbola is in the 2nd and 4th quadrant.