Final answer:
The question involves rotating a conic section equation to standard position, specifically focusing on eliminating the xy-term through a calculation of the rotation angle and applying it to obtain a new set of coordinates.
Step-by-step explanation:
The student is asking about rotating a conic section given by the equation 5x² + 6xy + 5y² - 9x + 4y = -2 into standard position. Rotating the conic involves removing the xy-term, which is done through a coordinate rotation. However, before we proceed, it's necessary to point out that the provided question contains several typos and irrelevant sections that are not connected to the task of rotating a conic section. Therefore, they will be ignored in the solution process. The rotation angle θ is calculated using the formula θ = 1/2 · arctan(2B / (A - C)), where A, B, and C represent the coefficients of the quadratic part of the equation. For this example, A = 5, B = 3, C = 5. After calculating θ, a new set of coordinates (x', y') are defined through the rotation, which eliminates the xy-term and provides the equation in standard form.