Key Concept: The Standard Equation of a Hyperbola and its Features

A hyperbola centered at the origin, with its transverse axis lying along the x-axis (meaning its foci and vertices are on the x-axis), has the standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here, $a$ and $b$ are positive real numbers representing the lengths of the semi-transverse and semi-conjugate axes, respectively. From this standard form, we can define its key features:

• Semi-transverse axis ($a$): The distance from the center to a vertex.
• Vertices: Located at $(\pm a,