Determine if the transverse axis is horizontal or vertical for the following hyperbolas.
$3x^2-y^2=18$

• A. Horizontal
• B. Vertical
• C. Cannot be determined

Answer: (A)

Verified step by step guidance

1. Rewrite the given equation in the standard form of a hyperbola by dividing both sides of the equation by 18: \(\frac{3x^2}{18} - \frac{y^2}{18} = \frac{18}{18}\).
2. Simplify the fractions to get: \(\frac{x^2}{6} - \frac{y^2}{18} = 1\).
3. 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 \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
4. 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.
5. Conclude that since the hyperbola matches the first form, the transverse axis is horizontal.