Final answer:
To diagonalize the conic section given by the equation 2x² + 4xy - y² = 24, one must apply a rotation transformation to find the angle θ that eliminates the xy-term. Following this, the conic can be expressed in new coordinates aligned with the principal axes, without the xy term, thus completing the process of diagonalization.
Step-by-step explanation:
Diagonalization of a Conic Section
To rotate the conic section 2x² + 4xy - y² = 24 to principal axes, we need to remove the xy-term which signifies the rotation of the axes. This process, known as diagonalization, involves finding a new set of axes on which the conic can be expressed without the xy-term. To do this, we look for an angle θ that will eliminate the xy-term via a rotation transformation.
The transformed equation can be found by using the rotation formulas x = x'cos(θ) - y'sin(θ) and y = x'sin(θ) + y'cos(θ). Substituting these and comparing coefficients will allow us to solve for the angle θ that removes the xy term. Once θ is found, we can write the equation purely in terms of x' and y', which will be aligned along the major and minor axes of the conic respectively.
In this specific case, if we substitute and compare coefficients, we would find that tan(2θ) = -4/(2x² - y²), which gives us the information needed to solve for θ. We then use this angle to express our quadratic form in terms of the new primed coordinates without the xy term, achieving the goal of diagonalization and revealing the conic's principal axes.
Steps for Diagonalization
Write down the given conic equation.
- Apply the rotation transformation formulas to the x and y variables.
- Find the angle θ that results in the elimination of the xy-term.
- Substitute this angle back and rewrite the equation in the transformed coordinates.