Problem 72 - College Algebra

Question

How can you distinguish an ellipse from a hyperbola by looking at their equations?

Step-by-Step Solution

Answer

By examining the standard forms of their equations, an ellipse is characterized by the addition of the squared terms while a hyperbola has a subtraction sign between them.

Step 1: Standard Form of Ellipse

The general form of an ellipse with center at origin and axes along the x and y axes is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The coefficients of the squared terms are both positive and the terms are combined by addition. Note, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

Step 2: Standard Form of Hyperbola

The general form of a hyperbola with center at origin and axes along the x and y axes is given by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\). Notice one key difference-- the terms are separated by a subtraction sign (not addition). These equations represent hyperbolas opening left-right and up-down, respectively.

Step 3: Distinguishing Ellipse from Hyperbola

By comparing the standard forms of ellipse and hyperbola, the rule of thumb to distinguish them is as follows: An ellipse has both squared terms added together (both terms have positive coefficients), while a hyperbola has a minus sign between the squared terms (one term has a negative coefficient).