Final answer:
To find the complex number pairs (x, y) that with a and b form a square, rotate the side ab by 90 degrees to get bx and ay. Solve x = b + i*(a - b) and y = a + i*(b - a) to find the vertices of the square.
Step-by-step explanation:
To find all pairs of complex numbers (x, y) such that a, b, x, y are vertices of a square on the complex plane with ab, bx, xy, ya as its sides, you must utilize the properties of squares and complex numbers.
A square in the complex plane has sides of equal length and the adjacent sides are orthogonal (form a right angle to each other).
Let's represent the complex numbers a and b as a = a1 + ib1 and b = a2 + ib2. The side ab is a complex vector from a to b, and bx would be the side of length equal to ab, but rotated 90 degrees in the complex plane. To achieve this, you can multiply the vector representing ab by i (i.e., 0 + i1), which represents a 90-degree rotation in the complex plane.
Thus, to find x, use the equation x = b + i*(a - b). Similarly, to find y, you would take a and add the rotated vector ab, resulting in y = a + i*(b - a). You now have the required complex numbers x and y which, along with a and b, form the vertices of a square on the complex plane.