1.

Let bar z+b bar z= c, b ne 0 be a line in the complex plane where bar b

Answer»

SOLUTION :LetQ be `z_2` and its reflection be the p `(z_1)` in the given line . If O(z) be any point on the given line then by definition OR is right bisector of QP.
`therefore ""Op=OQ or |z-z_1|=|z-z_2|`
`rArr|z-z_1|^2=|z-z_2|^2`
`rArr(z-z_1)(barz-barz_1)=(z - z_2)(barz-barz_2)`
`rArr(barz_1-barz_2)+bar z(z_1 -z_2)=z_1 bar z_1- z_2barz_2`
Comparing with given line `zbarb + bar ZB = C`
`(barz_1-barz_2)/(BARB)=(z_1-z_2)/(B)=(z_1barz_1-z_2barz_2)/c=lambda`
`(barz_1-barz_2)/(lambda)=bar b ,(z_1-z_2)/(lambda)b,(z_1barz_1-z_2 barz_2)/(lambda)c ......(i) `
`therefore barz_1 b+z_2barb=barz_1((z_1-z_2)/(lambda))+z_2((barz_1-barz_2)/(lambda))`
`=(z_1barz_1-z_2barz_2)/(lambda)=c """[from Eq. (i)]"`


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