Ohio Assessments for Educators (OAE) Mathematics Practice Exam

In a hyperbola, what is the standard form equation?

1. (x-h)²/a² - (y-k)²/b² = 1

2. (y-k)²/a² - (x-h)²/b² = 1

3. (x-h)² + (y-k)² = r²

4. (x-h)² - (y-k)² = 1

The correct answer is: (x-h)²/a² - (y-k)²/b² = 1

The standard form of the equation for a hyperbola is given by the expression that distinctly showcases the difference in squares of the variables. The correct form, which highlights the orientation of the hyperbola, is represented by the equation where one variable is subtracted from the other, indicating that the hyperbola opens along the x-axis. In this case, the equation (x-h)²/a² - (y-k)²/b² = 1 illustrates that the center of the hyperbola is located at the point (h, k), with 'a' and 'b' representing the distance from the center to the vertices along the x-axis and y-axis, respectively. This structure helps to identify the principal components of a hyperbola, determining the asymptotes and the direction in which the hyperbola opens. The other formulations either do not conform to the standard requirements of a hyperbola or represent different conic sections. For instance, another equation may represent a circle or an ellipse, or may not convey the proper relationship between the variables as required for hyperbolas. Thus, the first option accurately encapsulates the defining features of a hyperbola and is recognized as the standard form.