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$P$ represents the variable complex number $z$.Find the locus of $P$,if $\mid 2z-3\mid=2$

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

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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 $Re(z)=a,Im(z)=b$
Step 1:
Let $z=x+iy$
$\mid 2z-3\mid=2$
$\mid 2(x+iy)-3\mid=2$
$\Rightarrow \mid (2x-3)+2yi\mid=2$
Step 2:
On squaring both sides
$\mid (2x-3)+2yi\mid^2=4$
Step 3:
The locus of $P$ is a circle with centre at $(\large\frac{3}{2}$$,0)$ .
The locus of $P$ is a circle with centre at $(\large\frac{3}{2}$$,0)$ and radius $1$unit. .
answered Jun 10, 2013 by sreemathi.v
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