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Suppose we have a square with vertices at $0,$ $-6 13i,$ $-19 7i,$ and $-13-6i.$ Suppose that we want to multiply these points by a single complex number $a bi$ to get a square with vertices $0,$ $8 2i,$ $6 10i,$ and $-2 8i.$ What is $(a,b)

Respuesta :

LammettHash

The number we're looking for, a + bi, is such that

(-6 + 13i) (a + bi) = 8 + 2i

which means

a + bi = (8 + 2i) / (-6 + 13i)

Simplify the right side :

a + bi = (8 + 2i) / (-6 + 13i) • (-6 - 13i) / (-6 - 13i)

… = -((8 + 2i) (6 + 13i)) / ((-6)² - (13i)²)

… = - (48 + 12i + 104i + 26i²) / (36 - 169i²)

… = - (48 - 26 + 116i) / (36 + 169)

… = - (22 + 116i) / 205

Then (a, b) = (-22/205, -116/205).

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