a. Rewrite the equation in a rotated x^' y^' -system without an x^' y^' -term. Use the appropria

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A. Rewrite the equation in a rotated x^' y^' -system without...

a. Rewrite the equation in a rotated $x^{\prime} y^{\prime}$ -system without an $x^{\prime} y^{\prime}$ -term. Use the appropriate rotation formulas from Exercises $7-18$ b. Express the equation involving $x^{\prime}$ and $y^{\prime}$ in the standard form of a conic section. c. Use the rotated system to graph the equation. $$3 x y-4 y^{2}+18=0$$

a. Rewrite the equation in a rotated $x^{\prime} y^{\prime}$ -system without an $x^{\prime} y^{\prime}$ -term. Use the appropriate rotation formulas from Exercises $7-18$
b. Express the equation involving $x^{\prime}$ and $y^{\prime}$ in the standard form of a conic section.
c. Use the rotated system to graph the equation.
$$3 x y-4 y^{2}+18=0$$

Key Concepts

Graphing Conic Sections

After rewriting the equation in the rotated coordinate system and expressing it in standard form, the next step is graphing. Graphing involves interpreting the parameters of the conic, such as its center, vertices, axes, and, if applicable, foci, to produce an accurate visual representation of the shape. The rotation might then be reversed to display the graph in the original coordinate system if needed.

Elimination of the Cross Term

In the context of conic sections, the elimination of the xy-term is crucial. By choosing an appropriate rotation angle ?, the mixed term can be removed from the equation. This process typically involves setting the coefficient of the new x'y'-term to zero, which leads to an equation that is easier to classify and analyze.

Standard Forms of Conic Sections

Conic sections—whether ellipses, hyperbolas, or parabolas—have standard forms that simplify identification, classification, and graphing. Once the equation is rewritten without the cross term, it can usually be rearranged into one of these recognizable forms. This aids in determining key features like vertices, axes, and foci.

Rotation of Axes

This concept involves rotating the coordinate system by a certain angle so that the given equation can be expressed without the xy-process term (or cross term). The fundamental idea is to simplify the equation by aligning the new axes with the inherent symmetry of the conic section, which often eliminates the mixed term and makes further analysis easier.

Transformation Formulas

Transformation formulas relate the original coordinates to the rotated coordinates. They take the form x = x'cos? - y'sin? and y = x'sin? + y'cos?. These formulas are essential for rewriting equations in the rotated coordinate system, ensuring that every term is correctly transformed.

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