Problem 72 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The equations of an ellipse and a hyperbola can be distinguished by looking at the signs of the terms. In ellipse equation all terms are added together, whereas in a hyperbola equation one term is subtracted from the other.

Step 1: Understand the Equations

The general form of equation for an ellipse in standard form is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) and for a hyperbola it is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \). The variables \( h \) and \( k \) are the coordinates of the center of the ellipse or the hyperbola. The variables \( a \) and \( b \) determine the shape of the ellipse and hyperbola.

Step 2: Identify the Difference

In an ellipse, all terms are added together while in a hyperbola, one term is subtracted from the other. This is the main distinguishing factor when comparing equations of an ellipse and a hyperbola.

Step 3: Summary

If all terms in the equation are positive, it defines an ellipse. If one term is subtracted from the other, it defines a hyperbola.