Problem 87 - College Algebra

Question

Will help you prepare for the material covered in the next section. Divide both sides of \(4 x^{2}-9 y^{2}=36\) by 36 and simplify. How does the simplified equation differ from that of an ellipse?

Step-by-Step Solution

Answer

After division and simplification, the equation transforms to \(x^{2}/9 - y^{2}/4 = 1\) which represents a hyperbola, not an ellipse due to the presence of a negative sign between the terms.

Step 1: Initial Set Up

The goal is to divide both sides of the given equation \(4 x^{2}-9 y^{2}=36\) by 36, resulting in a more simple equation.

Step 2: Divide Both Sides

Divide both sides of the original equation by 36. This yields \((4 x^{2})/36 - (9 y^{2})/36 = 36/36\).

Step 3: Simplify

Next is to simplify the above equation. Algebraically simplify to get \(x^{2}/9 - y^{2}/4 = 1\b).

Step 4: Comparison with Ellipse Equation

An equation of an ellipse usually comes in the form \[ (x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} =1 \]. Yet, the simplified equation has a term with a negative sign, which is against the form of the ellipse. Therefore, it differs because the sign between the terms in the equation indicates that this is a hyperbola and not an ellipse.