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