Determine the value of k such that the image of circle <mrow class="MJX-TeXAtom-ORD">

Sylvia Byrd

Sylvia Byrd

Answered question

2022-07-08

Determine the value of k such that the image of circle | z 1 | = k by complex function f ( z ) = z 3 1 2 z is a line.
I substitute z in first equation by f ( z ) then found that every k makes the equation in form of a z + b which is a linear equation. Is this true? But I really doubt. Help me

Answer & Explanation

Elijah Benjamin

Elijah Benjamin

Beginner2022-07-09Added 10 answers

f is a Möbius transformation, and therefore maps a circle to a circle or to a line. The image is a line if and only if some point of the circle is mapped to .
In your case f ( 1 2 ) = , so the image is a line if z = 1 2 is on the circle | z 1 | = k. That gives a unique value for k.

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