Question

For a complex number $Z$ , if the argument of $3+3i$ and $\left(Z - 2\right)\left(\overset{-}{Z} - 1\right)$ are equal, then the maximum distance of $Z$ from the $x$ -axis is equal to (where, $i^{2}=-1$ )

• A. $\frac{\left(1 + \sqrt{2}\right)}{2}$ units
• B. $2$ units
• C. $\frac{3}{2}$ units
• D. $\frac{\left(\sqrt{2} + 2\right)}{2}$ units

Answer 1

Answer: (A)

Let, $Z=x+iy$
Now, $\left(Z - 2\right)\left(\overset{-}{Z} - 1\right)=Z\overset{-}{Z}-2\overset{-}{Z}-Z+2$
$=x^{2}+y^{2}-3x+2+iy$
$arg$ of $\left(3 + 3 i\right)=\frac{\pi }{4}$
$\Rightarrow x^{2}+y^{2}-3x+2=y\left(where y > 0\right)$
$\Rightarrow x^{2}+y^{2}-3x-y+2=0$
Centre of the circle is $\left(\frac{3}{2} , \frac{1}{2}\right)$ and radius $=\frac{1}{\sqrt{2}}$

$\Rightarrow $ required value $=\frac{1}{2}+\frac{1}{\sqrt{2}}$ units