11 methods · 22 worked questions using these methods
★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.
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To identify the image curve of a constrained z under a map w=f(z), either separate w=u+iv into real and imaginary parts and eliminate the parameter, or carry z=reiθ and read off ∣w∣ and argw. A real or imaginary part fixed gives a line; a fixed modulus or a Mobius map of a line or circle gives a circle or line.
Trigger: "find / sketch the locus of w" where w is a given function of a z that runs along a line, a circle or an arc.
Instances: (i) Split into u,v and eliminate: lines u= const or v= const map to identifiable curves (often after a tan(θ/2) substitution). (ii) Mobius map w=cz+daz+b sends lines and circles to lines and circles; test a few points or use the circle equation. (iii) Translate an argument condition argz−z2z−z1=α into an arc of a circle, and a modulus condition z−z2z−z1=k into a circle or line.
