Final answer:
To find the equations of conics relative to principal axes, we need to complete the square. In this case, the given equation represents an ellipse. By rearranging and factoring, we can find the equation of the ellipse relative to the principal axes.
Step-by-step explanation:
To find the equations of conics relative to principal axes, we need to complete the square. When we have a quadratic equation in the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, the coefficients A, B, and C determine the type of conic section. In this case, A = 3, B = 8, and C = -3.
The equation 3x² + 8xy - 3y² = 8 represents an ellipse. To find the equation of the ellipse relative to the principal axes, we can complete the square. Rearranging the equation, we get 3x² + 8xy - 3y² - 8 = 0. Factoring the coefficients of x² and y², we have (x+y)² - 16y² - 8 = 0.
By dividing the equation by the constant term, we get ((x+y)/√8)² - (2y/√8)² - 1 = 0. Simplifying further, we have (x + y)/√8)² - (y/√2)² - 1 = 0. This represents an ellipse with the equation (x+y)/√8)² - (y/√2)² = 1.