Area of a Parallelogram Graphed on Real and Complex Plane

Algebra Level 3

The Argand Diagram is a method of graphing points similar to the Cartesian coordinate system. Instead, however, points on the Argand Diagram, also called the complex coordinate plane, are represented as complex numbers in the form \(a+bi\), where the real component \(a\) represents the \(x\)-coordinate, and the imaginary component \(b\) represents the \(y\)-coordinate. When \(f(x)=-x^{2}+6x-45\) is graphed on the Cartesian coordinate plane, let the two roots be \(z_1\) and \(z_2\). The points \((8, 13)\) and \((8, 1)\) are graphed on the Cartesian coordinate plane, and \(z_1\) and \(z_2\) are graphed on the Argand Diagram. Then, they two systems are overlapped such that their origins are equal and they are proportionate (i.e. the point \(3i\) on the Argand Diagram is equal\((0, 3)\) on the Cartesian coordinate plane). What is the area of the parallelogram formed by the four points?

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