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Chapter 6: Problem 18

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. $$6 y^{2}-3 x^{2}=18$$

Short Answer

Expert verified
The center is (0,0), vertices are at (0,卤鈭�3), the foci are at (0,卤3), and the equations of the asymptotes are y = 卤鈭歕(\frac{1}{2}\)x.

Step by step solution

01

Getting equation in standard form

First arrange the given equation into the standard form of the hyperbola equation. In this case, the hyperbola opens upwards and downwards because \(y^{2}\) is positive. Rearrange the given equation \(6y^{2} - 3x^{2} = 18\) as: \(\frac{{y^{2}}}{3} - \frac{{x^{2}}}{6} = 1\). Here, \(a^{2} = 3\) and \(b^{2} = 6\)
02

Center, Vertices, and Foci

The center of the hyperbola is where the graph intersects the x and y axis. The center is (0,0) because there is no added or subtracted quantity in x or y terms. \nThe vertices are (0, 卤a) = (0, 卤鈭�3).\nThe foci are located at (0, 卤c), where \(c^{2} = a^{2} + b^{2} = 3 + 6 = 9\) and hence c = 卤鈭�9, so the foci are (0,卤3)
03

Equations of the asymptotes

The equations of the asymptotes are given by \(y = 卤 \frac{a}{b}x\), here a = 鈭�3 and b = 鈭�6, so the equations of asymptotes are \(y = 卤 \frac{\sqrt{3}}{\sqrt{6}}x = 卤 \sqrt{\frac{1}{2}}x\). Hence, the equations of the asymptotes are \(y = 卤鈭歕frac{1}{2}x\)
04

Graphing the hyperbola

Use a graphing utility to draw the hyperbola and its asymptotes. Along the y-axis, plot points at 卤鈭�3 and 卤3 for the vertices and foci respectively. Also, plot the lines for the asymptotes using the slope 鈭歕(\frac{1}{2}\). The hyperbola opens upwards and downwards along these asymptotes.

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