What formula indicates the presence of a hyperbola?

The formula that indicates the presence of a hyperbola is indeed represented by the equation ( x^2/a^2 - y^2/b^2 = 1 ). This specific form signifies that the hyperbola opens along the x-axis with its center at the origin.

In hyperbolic geometry, the equation defines two separate curves that are symmetric about the x-axis, indicating that as the value of ( x ) increases or decreases, the value of ( y ) can take on real values that will maintain the equality of the equation. The coefficients ( a ) and ( b ) are real numbers that determine the shape and size of the hyperbola. Specifically, ( a ) corresponds to the distance from the center to the vertices along the x-axis, while ( b ) relates to the distance to the points that define the asymptotes.

Other forms mentioned in the options are characteristic of different conic sections. The equation ( x^2 + y^2 = r^2 ) describes a circle, which has consistent radii from the center in all directions. Meanwhile, ( x^2/a^2 + y^2/b^2 = 1 ) represents an ellipse, characterized by its