Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

The key difference between an ellipse and a hyperbola when looking at their equations is the sign between the two terms. An ellipse always has a positive (+) sign, while a hyperbola always has a negative (-) sign.

Step 1: Standard Equations of an Ellipse and Hyperbola

The general equation of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a > 0\) and \(b > 0\). The standard equation of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Step 2: Identify the Distinguishing Feature

Having looked at the equations, it can be observed that the distinguishing feature between an ellipse and a hyperbola is the sign between the two terms of the equation. In the ellipse, the sign is positive (+) whereas, in the hyperbola, it is negative (-). So, this is the key feature when distinguishing an ellipse from a hyperbola by using their equations.