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Please-how-did-z-a-r-became-z-a-re-i-




Question Number 195848 by pete last updated on 11/Aug/23
Please how did ∣z−a∣=r became   z= a + re^(iθ) ?
Pleasehowdidza∣=rbecamez=a+reiθ?
Answered by Frix last updated on 11/Aug/23
∣z−a∣=r ⇒ z−a=re^(iθ)   This step is always true, even for z−a∈R  because we can always find a θ: if z−a>0  ⇒ θ=0, if z−a<0 ⇒ θ=π  z−a=re^(iθ)  ⇔ z=a+re^(iθ)
za∣=rza=reiθThisstepisalwaystrue,evenforzaRbecausewecanalwaysfindaθ:ifza>0θ=0,ifza<0θ=πza=reiθz=a+reiθ
Answered by AST last updated on 11/Aug/23
∣z−a∣=r is the equation for the circle centered  at a with radius r  So,we get z(points on the circle) by translating  a circle with centre at the origin(0,0) and radius  r to a  re^(iθ)  ⇒the locus of points generated by rotating  a vector with magnitude r around the origin(this  gives a circle)  +a⇒translating re^(iθ)   by a,the centre(origin) is  also translated by a  ⇒re^(iθ) +a gives a circle with centre at a[∣z−a∣=r]
za∣=ristheequationforthecirclecenteredatawithradiusrSo,wegetz(pointsonthecircle)bytranslatingacirclewithcentreattheorigin(0,0)andradiusrtoareiθthelocusofpointsgeneratedbyrotatingavectorwithmagnituderaroundtheorigin(thisgivesacircle)+atranslatingreiθbya,thecentre(origin)isalsotranslatedbyareiθ+agivesacirclewithcentreata[za∣=r]

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