In Exercises 27–32, find the standard form of the equation of each hyperbola.

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola centered at the origin is given by the equations (x²/a²) - (y²/b²) = 1 or (y²/b²) - (x²/a²) = 1. Here, 'a' represents the distance from the center to the vertices along the x-axis or y-axis, while 'b' represents the distance to the asymptotes. Understanding this form is crucial for identifying the hyperbola's orientation and dimensions.

Asymptotes of a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola in standard form, the equations of the asymptotes can be derived from the standard form equations. They are given by y = (b/a)x and y = -(b/a)x for hyperbolas centered at the origin. Recognizing the asymptotes helps in sketching the hyperbola accurately and understanding its behavior.

Vertices and Co-vertices

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis, while the co-vertices are associated with the conjugate axis. For a hyperbola in standard form, the vertices are located at (±a, 0) or (0, ±b) depending on the orientation. Identifying these points is essential for graphing the hyperbola and understanding its geometric properties.