Determine if the transverse axis is horizontal or vertical for the following hyperbolas.
A
Horizontal
B
Vertical
C
Cannot be determined
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1
Rewrite the given equation in the standard form of a hyperbola by dividing both sides of the equation by 18: \(\frac{3x^2}{18} - \frac{y^2}{18} = \frac{18}{18}\).
Simplify the fractions to get: \(\frac{x^2}{6} - \frac{y^2}{18} = 1\).
Identify the form of the hyperbola. Since the \(x^2\) term is positive and comes first, and the \(y^2\) term is subtracted, the hyperbola is of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Recall that for hyperbolas in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the transverse axis is horizontal, while for \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), it is vertical.
Conclude that since the hyperbola matches the first form, the transverse axis is horizontal.