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I have a problem:

"A function $f$ is defined on the complex numbers by $f(z) = (a+bi)z$, where $a$ and $b$ are positive 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 $\mid a+bi\mid=8$, find the value of $b^2$."

What does "image of each point" mean in this context? The image of a point is generally defined with a certain line of reflection, but there seems to be no line of reflection defined. That's my only question, and I have no queries about the rest of the problem, but maybe there's some other context that I'm not noticing?

mpnm
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2 Answers2

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It means that $|z| = |f(z)-z|$. Said another way, this function ensures that $0$, $z$ and $f(z)$ always form an isosceles triangle in the complex plane.

CyclotomicField
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If $z$ is in the domain (here $\mathrm C$), then its image by the function $f$ is given by $f(z).$

The map need not be a reflection. It can be any map. And the value at $z$ is usually also called the image of $z$ by $f.$

Allawonder
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