Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the standard form of an ellipse centered at (h, k)?

1. (x-h)²/a² + (y-k)²/b² = 1

2. (x-h)² - (y-k)²/b² = 1

3. (y-k)²/a² + (x-h)²/b² = 1

4. (x-h)² + (y-k)² = r²

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

The standard form of an ellipse centered at the point (h, k) is expressed as (x - h)²/a² + (y - k)²/b² = 1, where 'a' represents the semi-major axis length and 'b' represents the semi-minor axis length. This particular form is specifically structured to reflect the geometry of an ellipse, which is defined as the set of points where the sum of the distances from two fixed points (the foci) remains constant. By centering the ellipse at (h, k), the equation clearly shifts the origin from (0, 0) to (h, k). The terms (x - h) and (y - k) indicate that the coordinates of points on the ellipse are measured relative to this new center. Furthermore, the presence of 'a²' and 'b²' in the denominators of the respective squared terms indicates that the ellipse stretches further along one axis than the other, which is a distinguishing factor compared to other conic sections. The equation equals 1, which is a standard requirement for this type of conic, ensuring that all points (x, y) satisfying this equation define an ellipse. The other choices present alternative forms