Multiple Choice Question (MCQ)

If z_1, z_2,z_3\in C, then z_1=z_2=z_3 is the for the equation (z_1-z_2)^2+(z_2-z_3)^2=0 to hold true.

  1. sufficient but not necessary condition

  2. ×

    necessary but not sufficient condition

  3. ×

    sufficient and necessary condition

  4. ×

    neither sufficient nor necessary condition

Collected in the board: Imaginary number

Steven Zheng posted 1 week ago

Answer

  1. If z_1=z_2=z_3, then (z_1-z_2)^2+(z_2-z_3)^2=0. So it's true z_1=z_2=z_3 is the sufficient condition for the equation to hold true. However, if z_1 = i, z_2=0 and z_3=1, then

    (i-1)^2+(0-1)^2=i^2+1^2 = -1+1=0

    which shows the equation is also equal to 0. So (z_1-z_2)^2+(z_2-z_3)^2=0 must not deduce the result of z_1=z_2=z_3. z_1=z_2=z_3 is not the necessary condition for the equation to hold true.

Steven Zheng posted 1 week ago

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