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Chapter 6: Problem 35

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. $$4 x^{2}-9 y^{2}=36$$

Short Answer

Expert verified
The center of the hyperbola is at (0,0), the vertices are at (\(\pm 2, 0\)), the foci are at (\(\pm \sqrt{13}, 0\)), and the equations of the asymptotes are \(y = \pm \frac{3}{2}x\).

Step by step solution

01

Identify the center

Since our equation is already centered at the origin (0, 0), the center of the hyperbola is at (0,0).
02

Identify the vertices

To find the vertices, we take the square root of the number under \(x^2\), which gives us the semi-major axis \(a\). The square root of 4 is 2. Thus, the vertices are at \((\pm 2, 0)\).
03

Identify the foci

To find the foci, we need to calculate \(c\), where \(c^2 = a^2 + b^2\). First, we find \(b\), which is the square root of the number under \(y^2\). The square root of 9 is 3, giving us \(b = 3\). Hence, \(c^2 = a^2 + b^2 = 2^2 + 3^2 = 4+9 = 13\), and \(c=\sqrt{13}\). Therefore, the foci are at \((\pm \sqrt{13}, 0)\).
04

Identify the equations of the asymptotes

The equations of the asymptotes are \(y = \pm \frac{b}{a}x\). Substituting \(a = 2\) and \(b = 3\), we get \(y = \pm \frac{3}{2}x\). Therefore, the equations of the asymptotes are \(y = \frac{3}{2}x\) and \(y = -\frac{3}{2}x\).
05

Graph the hyperbola and its asymptotes

Use a graphing utility to plot the hyperbola and its asymptotes. The hyperbola should open along the x-axis and will have a center at the origin, vertices at (\(\pm 2, 0\)), foci at (\(\pm \sqrt{13}, 0\)), and asymptotes with the equations \(y = \pm \frac{3}{2}x\).

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