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# A function $f$ is defined on the complex numbers by $$f(z) = (a + bi)z,$$where $a$ and $b$ are positive real numbers. This function has the

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A function $$f$$ is defined on the complex numbers by

$$f(z) = (a + bi)z,$$

where  $$a$$ and $$b$$ are positive real numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that

$$|a + bi| = 3$$

find $$a$$ and $$b$$

Dec 20, 2018

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$$a=\dfrac 1 2\\ b = \dfrac{\sqrt{35}}{2}$$

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Dec 20, 2018
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can i see the steps please?

RektTheNoob  Dec 21, 2018
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I'll give you a hint.

$$\text{since }f(z) \text{ has the stated property }\forall z\\ f(1) \text{ is equidistant from }(0,0i), \text{ and }(1,0i)\\ \text{The locus of these points is }\left(\dfrac 1 2 , i y\right),~\forall y \in \mathbb{R}\\ \text{From this you can figure out }a\\ \text{Once you have }a \text{ finding }b \text{ is trivial}$$

Rom  Dec 21, 2018
edited by Rom  Dec 21, 2018