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Chapter 6: Problem 34

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
Center: (0, 0), Vertices: (0, 鈭�3) and (0, -鈭�3), Foci: (0, 3) and (0, -3), Equations of the Asymptotes: \( y = 卤\frac{鈭�3}{鈭�6}x \)

Step by step solution

01

Identify a and b from the equation

Divide the equation \( 6y^2 - 3x^2 = 18 \) by 18 to convert it to the standard form. Now, the equation becomes \( \frac{y^2}{3} - \frac{x^2}{6} = 1 \). Therefore, \( a^2 = 3 \) and \( b^2 = 6 \). Taking square root, a = 鈭�3 and b = 鈭�6.
02

Calculate Center, Vertices, and Foci

For a hyperbola centered at origin, the center is at (0,0). The vertices are at (0, a) and (0, -a), substituting a = 鈭�3 gives vertices at (0, 鈭�3) and (0, -鈭�3). The foci are at (0, c) and (0, -c) where \( c = \sqrt{a^2 + b^2} \). Substituting a = 鈭�3 and b = 鈭�6 gives \( c = \sqrt{3+6} = \sqrt{9} = 3 \). So, the foci are at (0, 3) and (0, -3).
03

Find the Equations of the Asymptotes

The equations of the asymptotes for a hyperbola of the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) are \( y = \frac{a}{b}x \) and \( y = -\frac{a}{b}x \). Substituting a = 鈭�3 and b = 鈭�6 gives \( y = 卤\frac{鈭�3}{鈭�6}x \)
04

Graph the Hyperbola and Asymptotes

Plot the center, vertices and foci points on a graph. Sketch the hyperbola making sure it passes through the vertices and approaches the asymptotes as it extends. Draw the asymptote lines using the equations obtained in Step 3. Use a graphing calculator or online plotter for accuracy.

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