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Which standard equation represents a hyperbola?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, a hyperbola centered at the origin of a Cartesian coordinate system with its transverse axis lying on the x-axis is represented by the following standard equation:

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

In this equation, a and b are constants representing the lengths of the semi-axes of the hyperbola. This curve is a type of conic section defined as the locus of points where the difference of the distances from two fixed points, known as foci, remains constant.1