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Let z and z_(0)be two complex numbers. It is given that |z|=1and thatnumbersz,z_(0),zbarz_(0) 1, and0 arerepresented in a Argand diagram by thepoints P,P_(0),Q,Aandthe origin respectively. Showthat thetrianglesPOP_(0) and AOQ are congruent . Hence, orotherwise, prove that|z-z_(0)|= |z barz_(0) -1| |
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Answer» SOLUTION :Given `OA = 1 and |z|=1` `therefore OP=|z-0| =|z| =1` `rArr OP = OA` `OP_(0) =|z_(0) -0| =|z_(0)|` `OQ =|zz_(0)-0|` `=|zz_(0)| =|z||z_(0)| =|z_(0)|` Also, `angle P_(0)OP = ARG((z_(0)-0)/(z-0))` `= arg((z_(0))/(z)) = arg ((zbarz_(0))/(zbarz))` `=-arg((zbarz_(0))/(1)) = -argbar((barzz_(0)))` `= -arg(zbarz_(0)) = arg((1)/(zbarz_(0)))` `= arg((1-0)/(zbarz_(0) -0))` `angle AOQ` Thus, the triangle `POP_(0)` and AOQ are congurent. HENCE `PP_(0) = AQ rArr |z-z_(0)| = |zbarz_(0) -1|`
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