Question

Consider two complex number $z$ and $\omega$ satisfying $|z|=1$ and $|\omega -2|+|\omega -4|=2$. Then which of the following statement(s) is/are correct ?

• A. $\text{Re}\left(z\omega \right)$ can never exceed $4$.
• B. If $\arg z=\arg \omega$, then $|z+2|$ is equal to $3$.
• C. The minimum value of $|z-\omega |$ is equal to $2$.
• D. The maximum value of $|z-w|$ is equal to $6$.

Answer 1

Answer: (A), (B)

The correct options are
A $\text{Re}\left(z\omega \right)$ can never exceed $4$.
B If $\arg z=\arg \omega$, then $|z+2|$ is equal to $3$.

$z$ lies on the unit circle centred at the Origin.

& $\omega$ lies on the line segment joining $C(2,0)$ & $D(4,0)$



$|z-\omega {|}_{\text{min}}=BC=1$
$|z-\omega {|}_{\text{max}}=AD=5$

$z=e^{i\theta }\Rightarrow z\omega =\omega e^{i\theta }$
$\Rightarrow \text{Re}\left(z\omega \right)=\omega \cos \theta \le \omega \le 4$

$\arg \left(\omega \right)=0\Rightarrow \arg \left(z\right)=0$
So, $z$ lies at $B(1,0)$
So, $|z+2|=|1+2|=3$.