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Let z,z(0) be two complex numbers. It is...

Let `z,z_(0)` be two complex numbers. It is given that `abs(z)=1` and the numbers `z,z_(0),zbar_(0),1` and 0 are represented in an Argand diagram by the points P,`P_(0)`,Q,A and the origin, respectively. Show that `/_\POP_(0)` and `/_\AOQ` are congruent. Hence, or otherwise, prove that
`abs(z-z_(0))=abs(zbar(z_(0))-1)=abs(zbar(z_(0))-1)`.

Text Solution

Verified by Experts

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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