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Q)

$P$ represents the variable complex number $z$.Find the locus of $P$,if $\mid z-5i\mid=\mid z+5i\mid$

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

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A)
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 $Re(z)=a,Im(z)=b$
  • 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:
$P$ represents the variable complex number $z$ on the complex plane where $\mid z-5i\mid=\mid z+5i\mid$
Let $z=x+iy$
$\mid x+iy-5i\mid=\mid x+iy+5i\mid$
$\mid x+i(y-5)\mid^2=\mid x+i(5+y)\mid^2$
Step 2:
Therefore $x^2+(y-5)^2=x^2+(y+5)^2$
$(y+5)^2-(y-5)^2=0$
$\Rightarrow 20y=0$
$y=0$
The locus of $P$ is the $x$-axis.
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