1.

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|

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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