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

step 1 From exercise 9 we have x is equal to square root of 2 divided by 2 multiplied by x -d s minus y

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. $$3 x^{2}-10 x y+3 y^{2}-32=0$$

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.
$$3 x^{2}-10 x y+3 y^{2}-32=0$$

Key Concepts

Rotation of Axes

This concept involves changing the coordinate system by rotating the axes in order to eliminate the mixed xy-term in a quadratic equation. It is particularly useful when dealing with conic sections whose equations include an xy-term, as the rotation simplifies the expression, facilitating the determination of the conic’s type and properties.

Coordinate Transformation Formulas

These formulas, such as x = x'cos? - y'sin? and y = x'sin? + y'cos?, are used to convert the coordinates from the original system to the rotated system. They provide the means to substitute the variables in the original equation, allowing the elimination of the xy-term when the appropriate rotation angle is chosen.

Standard Forms of Conic Sections

Once the equation of a conic is transformed to remove the cross-product term, it can often be rewritten in one of the standard forms that represent ellipses, hyperbolas, or parabolas. These forms clearly delineate the key features of the conic—such as center, axes, vertices, and foci—making it easier to classify and analyze.

Graphing Conic Sections

After rewriting a conic section in standard form, the next step involves graphing it based on the identified parameters like center, axes lengths, and orientation. This approach uses the geometric properties derived from the standard form to accurately represent the conic's shape and position in the coordinate plane.

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