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Chapter 6: Problem 17

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. $$2 x^{2}-3 y^{2}=6$$

Short Answer

Expert verified
The center of the hyperbola is (0,0). The vertices are at (\(\pm \sqrt{3}\),0), and the foci are at (\(\pm \sqrt{5}\), 0). The equations for the asymptotes are \(y = \pm \frac{\sqrt{2}}{\sqrt{3}}x\).\nThe graph would show a hyperbola opening along the x-axis with these detected vertices.

Step by step solution

01

Express the given equation in standard form

Rewrite the given equation \(2x^2 - 3y^2 = 6\) to the form of \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can divide by 6 to get: \(\frac{x^2}{3} - \frac{y^2}{2} = 1\) where \(a^2=3\) and \(b^2=2\).
02

Find the center, vertices, and foci of the hyperbola

The center of hyperbola is at the origin (0, 0). The vertices are at \((\pm a, 0)\) or \((\pm\sqrt{3}, 0)\). The foci are at \((\pm c, 0)\) where \(c=\sqrt{a^2 + b^2} = \sqrt{5}\). Therefore, the foci are \((\pm \sqrt{5}, 0)\).
03

Find the equations of the asymptotes

The equations of asymptotes are \(y = \pm \frac{b}{a}x =\pm \frac{\sqrt{2}}{\sqrt{3}}x\).
04

Graph the hyperbola and its asymptotes

Use a graphing utility and sketch the center, vertices and foci to visualize the hyperbola. Also plot the lines \(y = \frac{\sqrt{2}}{\sqrt{3}}x\) and \(y = -\frac{\sqrt{2}}{\sqrt{3}}x\) for the asymptotes.

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