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Chapter 6: Problem 33

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 at (0,0), the vertices are at (\(\pm\sqrt{3}\), 0), the foci are at (\(\pm\sqrt{5}\), 0), and the equations of the asymptotes are \(y = \pm \frac{\sqrt{2}}{\sqrt{3}}x\).

Step by step solution

01

Normalize the Hyperbola's Equation.

Start by re-writing the equation in standard form, like \( \frac{{(x-h)^2}}{{a^2}} - \frac{{(y-k)^2}}{{b^2}} = 1 \), for hyperbolas that open left and right or \( \frac{{(y-k)^2}}{{a^2}} - \frac{{(x-h)^2}}{{b^2}} = 1 \) for ones that open up and down. For this equation, divide both sides by 6 to get \(\frac{x^2}{3} - \frac{y^2}{2} = 1\). Here, \(a^2 = 3\), \(b^2 = 2\), and the center (h,k) is at the origin (0,0), because the x and y terms are not shifted.
02

Find the Vertices

The vertices are points where the hyperbola intersects its principal axis. For hyperbolas that open left and right, the vertices are at \((h \pm a, k)\) and for ones that open up and down the vertices are at \((h, k \pm a)\). For this hyperbola, the vertices are \(\pm \sqrt{a^2} = \pm \sqrt{3}\) along x-axis.
03

Find the Foci

The foci are found by calculating \(c\), where \(c^2 = a^2 + b^2\). For this hyperbola, \(c = \sqrt{a^2 + b^2} = \sqrt{3 + 2} = \sqrt{5}\). The foci are at \((h \pm c, k)\) for hyperbolas that open left and right and at \((h, k \pm c)\) for hyperbolas that open up and down. Here, the foci are located at \((\pm\sqrt{5}, 0)\).
04

Find the Equations of the Asymptotes

The asymptotes of the hyperbola are given by the equations \( y = k \pm \frac{b}{a}(x - h) \). For this hyperbola, the equations are \( y = \pm \frac{\sqrt{2}}{\sqrt{3}}x \).
05

Graph the Hyperbola

Using a graphing utility, plot the hyperbola and its asymptotes. Remember that the hyperbola's center, foci, and vertices are located at the origin, \((\pm \sqrt{5},0)\), and \((\pm\sqrt{3}, 0)\) respectively, and its asymptotes have equations \(y = \pm \frac{\sqrt{2}}{\sqrt{3}}x\).

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