Final answer:
The transformation of the given quadratic equation involves shifting the origin to (-2, 2) and applying a 45° rotation of axes, using trigonometric functions to express the new coordinates in terms of the old system.
Step-by-step explanation:
To transform the given equation x² + 3xy + y² – 2x + 2y - 1 = 0 to rectangular axes with the origin shifted at (-2, 2) and the axes rotated by 45°, we need to perform a coordinate transformation. First, we apply the translation of the origin by using the new coordinates (x', y') where x = x' - 2 and y = y' + 2. After substituting these into the original equation and simplifying, we then need to rotate the axes by 45°. For a rotation by θ, the transformation equations are x' = x*cos(θ) - y*sin(θ) and y' = x*sin(θ) + y*cos(θ). However, since cos(45°) = sin(45°) = √2/2, we apply the specific rotation to get the final transformed equation in standard form.