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$P$ represents the variable complex number $z$.Find the locus of $P$,if $Im\bigg[\large\frac{2z+1}{iz+1}\bigg]$$=-2$

This is the first part of the multi-part Q8.

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1 Answer

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  • If $z=a+ib$ then $\bar{z}=a-ib$.
  • $\mid z\mid=\sqrt{a^2+b^2}$
  • $z^{-1}=\large\frac{a-ib}{a^2+b^2}$
  • $z\bar{z}=a^2+b^2$
  • Also $Re(z)=a,Im(z)=b$
  • If $z_1=a+ib,z_2=c+id$
  • $z_1z_2=(a+ib)(c+id)=(ac-bd)+i(ad+bc)$
  • $\mid z_1z_2\mid=\mid z_1\mid\mid z_2\mid$
Step 1:
Let $z=x+iy$
$\qquad\quad=\large\frac{2x+1)+2iy}{1-y)+ix}\times \big(\large\frac{(1-y)-ix}{(1-y)-ix}\big)$
Step 2:
The locus of $P$ is a straight line.
answered Jun 10, 2013 by sreemathi.v
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