Problem 49 - Algebra and Trigonometry

Question

Use a graphing utility to graph each equation. $$9 x^{2}+24 x y+16 y^{2}+90 x-130 y=0$$

Step-by-Step Solution

Answer

This equation is recognized as an ellipse. The graph is obtained using a graphing calculator and the key features such as the center, the lengths of the major and minor axes, and the orientation of the axes are identified.

Step 1: Recognize the Equation Type

Recognize that the equation \(9x^{2} + 24xy + 16y^{2} + 90x - 130y = 0\) is a conic section, and in particular it is an ellipse by its canonical form.

Step 2: Re-arrange the Equation

Re-arrange the equation and write it in general form for better visualization. The general form is \(Ax^{2} + 2Hxy + By^{2} + 2Gx + 2Fy + C = 0\).

Step 3: Use a Graphing Calculator

Plug the equation into a graphing calculator or a graphing software. Pay attention to determine the key features such as center, the lengths of the major and minor axes, and the orientation of the axes.

Step 4: Analyze the Graph

After graphing the equation, analyze the graph and identify key features of the ellipse which include the center, the lengths of the major and minor axes, and the orientation of the axes.