Which of the following is the standard equation of a hyperbola centered at the origin with a transverse axis along the x-axis?

• A. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
• B. $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
• C. $\frac{y^2}{x^2}-\frac{a^2}{b^2}=1$
• D. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Answer: (A)

Verified step by step guidance

1. Recall that the standard form of a hyperbola centered at the origin depends on the orientation of its transverse axis. If the transverse axis is along the x-axis, the equation has the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\).
2. Here, \(a^{2}\) and \(b^{2}\) are positive constants where \(a\) represents the distance from the center to each vertex along the transverse axis (x-axis), and \(b\) relates to the conjugate axis (y-axis).
3. If the transverse axis were along the y-axis, the equation would be \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\), which is not the case here.
4. The other options either have incorrect signs (such as addition instead of subtraction) or incorrect placement of variables and constants, which do not represent a hyperbola with a transverse axis along the x-axis.
5. Therefore, the correct standard equation for a hyperbola centered at the origin with a transverse axis along the x-axis is \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\).