Question

If `z_1, z_2, z_3` are distinct nonzero complex numbers and `a ,b , c in R^+` such that `a/(|z_1-z_2|)=b/(|z_2-z_3|)=c/(|z_3-z_1|)` Then find the value of `(a^2)/(|z_1-z_2|)+(b^2)/(|z_2-z_3|)+(c^2)/(|z_3-z_1|)`

Answer 1

`(a) /(|z_1 - z _2 |) = (b)/(| z_2 - z_3 |) = (c) /(|z_3 - z _1 |) = lamda ` (say )
` rArr a ^(2) = lamda ^(2) (z_1 - z_2 ) (bar z _1 - bar z _2 )`
` b^(2) = lamda ^(2) (z_2 - z_3 ) (bar z _2 - bar z _3 )`
` c ^(2) = lamda ^(2) (z _3 - z _ 1 ) ( bar z _3 - bar z_1 )`
` rArr (a^(2) )/(z _ 1 - z_ 2 ) + (b ^(2) ) /(z_ 2 - z_ 3 ) + (c ^(2))/(z_ 3 - z_ 1 ) = 0 `