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Complex numbers z_1,z_2,z_3 are the vertices A,B,C respectively of an isosceles right angled triagle with (z_1-z_2)^2=(z_1-z_3)(z_3-z_2) |
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Answer» Solution :SINCE , triangle is a RIGHT angled ISOSCELES triangle `therefore` Rotating `z_(2)` about `z_(3)` in anti-clockwise direction through an angle of `pi//2` , we get `(z_(2) - z_(3))/(z_(1) - z_(3)) = (|z_(2) - z_(3)|)/(|z_(1) - z_(3)|)e^((pi//2))` where , `|z_(2) = z_(3)| = |z_(1) - z_(3)|` `implies (z_(2) - z_(3))^(2) = -(z_(1) - z_(3))^(2)` `implies z_(2)^(2) + z_(3)^(2) - 2z_(2)z_(3) = - z_(1)^(2) - z_(3)^(2) + 2z_(1)z_(3)` `implies z_(1)^(2) + z_(2)^(2) - 2z_(1)z_(2) = 2z_(1)z_(3) + 2z_(2) z_(3) - 2 z_(3)^(2) - 2z_(1) z_(2)` `implies (z_(1) - z_(2))^(2) = 2{(2z_(1) z_(3)^(2)) + (z_(2)z_(3) - z_(1)z_(2))}` `implies (z_(1) - z_(2))^(2) = 2(z_(1) - z_(3)) (z_(3) - z_(2))`
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