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

step 1 From exercise 8 we have x is equal to square root of 2 multiplied by x -dess minus y -daz divide

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. $$x^{2}+4 x y+y^{2}-3=0$$

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.
$$x^{2}+4 x y+y^{2}-3=0$$

Key Concepts

Rotation of Axes

Rotation of axes refers to the method of changing the coordinate system to simplify the equation of a curve, commonly by eliminating the xy-term from a conic section’s equation. This transformation uses trigonometric formulas to substitute new variables (x' and y') in place of the old ones, thereby aligning the coordinate axes with the conic's principal directions, which often simplifies further analysis and graphing.

Elimination of the Cross-product Term

Eliminating the xy-term in a quadratic equation is a key step in simplifying the representation of a conic section so that it takes on a standard form. This is achieved by choosing an angle of rotation that nullifies the coefficient of the x’y’ term. The process often involves using specific trigonometric identities, such as tan 2? = B/(A – C), which determines the appropriate rotation angle.

Standard Forms of Conic Sections

Once the conic’s equation is free of the cross-product term, it can be manipulated into standard form. Standard forms clearly illustrate the essential geometric features of the conic—such as its center, axes, vertices, and foci—and allow for straightforward classification into ellipses, hyperbolas, or parabolas. This classification plays an important role in understanding the behavior and properties of conic sections.

Graphing Conic Sections

Graphing a conic section involves plotting the curve based on its standard form, which reveals the key characteristics such as vertices, axis of symmetry, and orientation. In contexts where the equation has been rotated, graphing in the new (x’, y’) system and then translating the graph back to the original coordinate system can provide a clearer picture of the conic's geometry and behavior.

Transcript

Please give Ace some feedback