Question

Find the regions of the z-plane for which $\left|\frac{z-a}{z+\overline{a}}\right|<1,=1$ or $>1$. when the real part of a is positive.

• A. The required regions are the right half of the z-pane, the imaginary axis and the left half of the z-plane respectively.
• B. The required regions are the left half of the z-pane, the imaginary axis and the right half of the z-plane respectively.
• C. The required regions are the right half of the z-pane the real axis and the left half of the z-plane respectively.
• D. The required regions are the left half of the z-pane the real axis and the right half of the z-plane respectively.

Answer 1

The correct option is A The required regions are the right half of the z-pane, the imaginary axis and the left half of the z-plane respectively.

Let
$z=x+iy$ and $a$ be a fixed complex number $(a+ib)$.
Hence
$|\frac{z-a-ib}{z+a-ib}|=1$
$|(x-a)+i(y-b)|=|(x+a)+i(y-b)|$
$(x-a{)}^2=(x+a{)}^2$
$2x\left(a\right)=0$
$x=0$ ... imaginary axis.
If
$|\frac{z-a-ib}{z+a-ib}|<1$
Then
$(x-a{)}^2<(x+a{)}^2$
Or
$x>0$ ... positive real axis.
If
$|\frac{z-a-ib}{z+a-ib}|>1$
$(x-a{)}^2>(x+a{)}^2$
$x<0$ ... negative real axis.