First slide

Algebra of complex numbers

Question

P(z) be a variable point in the Argand plane such that |z| = minimum {|z - 1|, |z + 1|}, then $z + z$ will be equal to

Moderate

A

- 1 or 1
B

1 but not equal to - 1
C

- 1 but not equal to 1
D

none of these

Solution

When |z - 1| < |z + 1| (or x > 0)
|z| = |z - 1|
$\begin{array}{l} \Rightarrow x^{2} + y^{2} = (x - 1)^{2} + y^{2} \\ \Rightarrow x = 1 / 2 \\ \Rightarrow z + z = 1 \end{array}$
When |z - 1| > |z + 1| (or x < 0),
|z| = |z + 1|
$\begin{array}{ll} \Rightarrow x^{2} + y^{2} = (x + 1)^{2} + y^{2} \\ \Rightarrow x = - 1 / 2 \\ \Rightarrow z + z = - 1 \end{array}$

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