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Chapter 11: Problem 11

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

Short Answer

Expert verified
Answer: The equations of the asymptotes for a standard hyperbola with vertices on the x-axis are \(y = \frac{b}{a}x\) and \(y = -\frac{b}{a}x\).

Step by step solution

01

Write the equation of a standard hyperbola with vertices on the x-axis

The equation of a standard hyperbola with vertices on the x-axis and center at the origin is given by: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] where \(a\) and \(b\) are the distances from the center to the vertices and co-vertices, respectively.
02

Find the slope of the asymptotes

The slope of the asymptotes of a hyperbola is determined by the ratio \(\frac{b}{a}\). The asymptotes will have the same magnitude and opposite sign slopes, meaning that their slopes will be \(\pm \frac{b}{a}\).
03

Write the equations of the asymptotes

Since the center of the hyperbola is at the origin, the asymptotes will pass through the origin (0, 0). We will use the point-slope form of a line to write the equation of these asymptotes: Asymptote 1 (positive slope): \(y = \frac{b}{a}x\) Asymptote 2 (negative slope): \(y = -\frac{b}{a}x\) These are the equations of the asymptotes for a standard hyperbola with vertices on the x-axis.

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