What are the equations for the asymptotes of this hyperbola?
[tex]$\frac{x^2}{9}-\frac{y^2}{64}=1$[/tex]
A. [tex]$y=\frac{3}{8} x, y=-\frac{3}{8} x$[/tex]
B. [tex]$y=\frac{\sqrt{55}}{5} x, y=-\frac{\sqrt{55}}{5} x$[/tex]
C. [tex]$y=\frac{\sqrt{73}}{5} x, y=-\frac{\sqrt{73}}{5} x$[/tex]
D. [tex]$y=\frac{8}{3} x, y=-\frac{8}{3} x$[/tex]
Answer (1)
Identify the values of a 2 and b 2 from the given hyperbola equation.
Calculate a and b by taking the square root of a 2 and b 2 .
Substitute the values of a and b into the asymptote equation y = ± a b x .
The equations of the asymptotes are y = 3 8 x , y = − 3 8 x .
Explanation
Identifying the Hyperbola Equation We are given the equation of a hyperbola: 9 x 2 − 64 y 2 = 1 We need to find the equations of its asymptotes.
Recalling Asymptote Equations The general form of a hyperbola centered at the origin is a 2 x 2 − b 2 y 2 = 1 The asymptotes of this hyperbola are given by the equations y = ± a b x
Finding a and b Comparing the given equation with the general form, we can identify the values of a 2 and b 2 : a 2 = 9 b 2 = 64
Calculating a and b Taking the square root of both sides, we find the values of a and b : a = 9 = 3 b = 64 = 8
Determining Asymptote Equations Now, we substitute the values of a and b into the equations of the asymptotes: y = ± 3 8 x This gives us two equations: y = 3 8 x y = − 3 8 x
Final Answer Therefore, the equations for the asymptotes of the given hyperbola are y = 3 8 x and y = − 3 8 x .
Examples
Understanding hyperbolas and their asymptotes is crucial in various fields. For instance, in physics, the trajectory of a spacecraft can sometimes be modeled as a hyperbola, and the asymptotes help predict its long-term path. Similarly, in economics, supply and demand curves can be modeled using hyperbolic functions, where asymptotes represent the theoretical limits of supply or demand as prices change. By understanding the behavior of hyperbolas, we can make informed predictions and decisions in these real-world scenarios.
