question_answer

Let \[z,{{z}_{0}}\]be two complex numbers \[{{\bar{z}}_{0}}\] being the conjugate of \[{{z}_{0}}\]. The numbers \[z,{{z}_{0}},z\,{{\bar{z}}_{0}}\] 1 and 0 \[P,{{P}_{0}},QA\] and the origin respectively if \[\left| z \right|=1,\] consider the following statement:

(I) \[PO{{P}_{0}}\] and \[AOQ\]are congruent

(II) \[\left| z-{{z}_{0}} \right|=\left| z{{{\bar{z}}}_{0}}-1 \right|\]

A) Only I is true

B) Only II is true

C) Both I & II are true

D) Both I & II are false

Correct Answer: C

Solution :

Given \[OA=1,OP=\left| z \right|=1\]

\[OA=OP\]

\[O{{P}_{0}}=\left| {{z}_{0}} \right|,OQ=\left| z{{{\bar{z}}}_{0}} \right|=\left| {{{\bar{z}}}_{0}} \right|=\left| {{z}_{0}} \right|..\left( 1 \right)\]

\[O{{P}_{0}}=OQ\]and\[\angle PO{{P}_{0}}=\arg \frac{{{z}_{0}}}{z}\]

\[\angle AOQ=\arg \frac{1}{z{{{\bar{z}}}_{0}}}=-\arg z{{\bar{z}}_{0}}\]\[=\arg \frac{{{z}_{0}}}{z}=\angle PO{{P}_{0}}\]

So triangles are congruent

\[P{{P}_{0}}=AQ\Rightarrow \left| z-{{z}_{0}} \right|=\left| z{{{\bar{z}}}_{0}}-1 \right|\]