1.

Letz_1, z_2 , z_3in Csuchthat|z_1 |= |z_2| =|z_3| =|z_1+z_2+ z _3|=4 . If|z_1 - z _2 |= | z _1+z _ 3 |andz_2nez_3, thenvaluesof|z_1+z_2 |* |z_1+z _ 3|is_____.

Answer»


Solution :Since , `|z_(1)| = |z_(2)| = |z_(3)| = 4`, oringin is circumcentre of triangleformed by `A(z_(1)),B(z_(2)) and C(z_(3))` .

So,orhtocentre of trinagle is `z_(1)+z_(2)+z_(3)|=4`
Given that `|z_(1)-z_(2)|=|z_(1)-z_(3)|`
`rArr AB = AC `
`therefore angle BAC = 90^(@)` (orthocentre is at vertex A)
Thus, `|z_(1)+z_(2)|.|z_(1)+z_(3)| = |z_(4) -z_(3)||z_(4)-z_(2)| `
`= (AC)(AB)= (4sqrt(2))(4sqrt(2)) = 32`


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