Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $|z|=\max \{|z-2|,|z+2|\}$, then
A
$|z+\bar{z}|=1$
B
$z+\bar{z}=2^2$
✓
$|z+\bar{z}|=2$
D
none of these
✓
Answer
Correct option: C.
$|z+\bar{z}|=2$
(C) $|z|=|z-2| \Rightarrow|z|^2=|z-2|^2$ $\Rightarrow z \overline{ z }=( z -2)(\overline{ z }-2)$ $\Rightarrow z \overline{ z }= z \overline{ z }-2 \overline{ z }-2 z +4$ $\Rightarrow z+\bar{z}=2$ ...(i) Also, $|z|=|z+2| \Rightarrow|z|^2=|z+2|^2$ $\Rightarrow z \overline{ z }=( z +2)(\overline{ z }+2)$ $=z \bar{z}+2(z+\bar{z})+4$ $\Rightarrow z+\bar{z}=-2$ ...(ii) From (i) and (ii), we get $|z+\bar{z}|=2$
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