For the hyperbola centered at the origin with equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, which expression gives the length of the transverse axis?

• A. $a$
• B. $2a$
• C. $2b$
• D. $\sqrt{a^2+b^2}$

Answer: (B)

Verified step by step guidance

1. Identify the standard form of the hyperbola equation given: \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\). This represents a hyperbola centered at the origin with the transverse axis along the x-axis.
2. Recall that for a hyperbola in this form, the transverse axis is the segment that passes through the two vertices on the x-axis, where the hyperbola intersects the x-axis.
3. The vertices of the hyperbola are located at \((\pm a, 0)\) because setting \(y=0\) in the equation gives \(\frac{x^{2}}{a^{2}} = 1\), so \(x = \pm a\).
4. The length of the transverse axis is the distance between these two vertices, which is the distance between \((-a, 0)\) and \((a, 0)\).
5. Calculate the length of the transverse axis as \$2a\(, since the distance between \)-a\( and \)a\( on the x-axis is \)a - (-a) = 2a$.