Final answer:
The question deals with diagonalizing a hyperbolic conic equation to find its principal axes through a rotation transformation.
Step-by-step explanation:
The question involves the use of diagonalization techniques to rotate the given conic section equation into its principal axes. The equation given is 2x² + 4xy - y² = 24. This equation represents a hyperbola, as can be inferred from the signs and coefficients of x² and y². To rotate this conic to its principal axes, we would typically complete the square for x and y, but in this case we need to eliminate the xy term by using a rotation of axes transformation.
The transformation involves substituting x and y with x' and y' where x = x' cos(θ) - y' sin(θ) and y = x' sin(θ) + y' cos(θ), and choosing θ such that the xy term is eliminated. Once this is done, we can identify the principal axes and the new representation of the conic section without the xy term, which simplifies the analysis and classification of the curve.