# $P$ 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.

Toolbox:
• 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
$\mid (2x-3)+2yi\mid^2=4$
$(2x-3)^2+(2y)^2=4$
$4x^2-12x+9+4y^2=4$
$4x^2+4y^2-12x+5=0$
$x^2+y^2-3x+\large\frac{5}{4}$$=0 Step 3: The locus of P is a circle with centre at (\large\frac{3}{2}$$,0)$ .
Radius=$\sqrt{\big(\large\frac{3}{2}\big)^2-\large\frac{5}{4}}$
$\quad\;\;\;\;\;=\sqrt{\large\frac{9}{4}-\frac{5}{4}}$$=1 The locus of P is a circle with centre at (\large\frac{3}{2}$$,0)$ and radius $1$unit. .