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N2010 P1 Q8

N2010 P1 Q8

Junior College 2
11 marks

The complex numbers \(z_1\) and \(z_2\) are given by \(1+\mathrm i\sqrt3\) and \(-1-\mathrm i\) respectively.

  1. Express each of \(z_1\) and \(z_2\) in polar form \(r(\cos\theta+\mathrm i\sin\theta)\), where \(r>0\) and \(-\pi<\theta\le\pi\). Give \(r\) and \(\theta\) in exact form.[2]
  2. Find the complex conjugate of \(\frac{z_1}{z_2}\) in exact polar form.[3]
  3. On a single Argand diagram, sketch the loci[4]
    1. \(|z-z_1|=2\),
    2. \(\arg(z-z_2)=\frac14\pi\).
  4. Find where the locus \(|z-z_1|=2\) meets the positive real axis.[2]

Solution:

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Answer:(i) \(r_1=2,\theta_1=\pi/3;r_2=\sqrt2,\theta_2=-3\pi/4\). (ii) \(\sqrt2(\cos(11\pi/12)+\mathrm i\sin(11\pi/12))\). (iv) \((2,0)\).

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