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Dear Pranay,Solution:- To prove- (z1 - z2)/(z¯1 - z¯2) + (z1 - z3)/(z¯1 - z¯3) = 0or, (z1 - z2)/(z1 - z3) + (z¯1 - z¯2)/(z¯1 - z¯3) = 0.......(1)From Rotation Th. of complex no.'s (Refer ?http://www.askiitians.com/iit-jee-algebra/complex-numbers/rotation.aspx), (z1 - z2)/(z1 - z3) = [|z1 - z2|/|z1 - z3|]exp(i∏/2) = i[|z1 - z2|/|z1 - z3|]Also, (z¯1 - z¯2) = conjugate of (z1 - z2) and |z¯1 - z¯2| = |z1 - z2| So, (z¯1 - z¯2)/(z¯1 - z¯3) = [|z1 - z2|/|z1 - z3|]exp(-i∏/2) = -i[|z1 - z2|/|z1 - z3|]Here, negative sign comes because of the conjugate (Refer ?http://www.askiitians.com/iit-jee-algebra/complex-numbers/conjugate-of-a-complex-number.aspx)Putting ?these values in eq. (1), we get-i[|z1 - z2|/|z1 - z3|] + (-i[|z1 - z2|/|z1 - z3|]) = 00 = 0Hence ProvedPlease feel free to post as many doubts on our discussion forum as you can. If you find any questionDifficult to understand - post it here and we will get you the answer and detailed solution very quickly. Weare all IITians and here to help you in your IIT JEE preparation.All the best Pranay!!!Regards,Priyansh BajajAskiitians Experts
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