Problem 11 - Calculus Early Transcendentals

Question

What are the equations of the asymptotes of a standard hyperbola with vertices on the \(x\) -axis?

Step-by-Step Solution

Answer

Answer: The equations of the asymptotes are \(y = \frac{b}{a}x\) and \(y = -\frac{b}{a}x\).

Step 1: Identify the general equation of the hyperbola

The general equation of a standard hyperbola with vertices on the x-axis is given by: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

Step 2: Find the slope of the asymptotes

The asymptotes of a hyperbola have the general form \(y = \pm \frac{b}{a}x\). The slope of the asymptotes is the ratio \(\frac{b}{a}\).

Step 3: Write the equations of the asymptotes

Using the slope \(\frac{b}{a}\), we can write the equations of the asymptotes as: \(y = \frac{b}{a}x\) and \(y = -\frac{b}{a}x\) These lines pass through the origin and have the slopes \(\frac{b}{a}\) and \(-\frac{b}{a}\), respectively.