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Find the locus of all complex numbers z satisfying the condition |z + 1/z| = |z - 1/z|.

1. y = x
2. y = -x
3. y = x, x not equal to 0
4. x² - y² = 0, x not equal to 0

Correct answer: x² - y² = 0, x not equal to 0

Solution

Setting z = x + iy gives 1/z = (x - iy)/(x² + y²). The condition |z + 1/z| = |z - 1/z| implies Re(z/conj(z)) type relation which simplifies to x² = y², i.e., y = plus or minus x (with z not zero). This is expressed as x² - y² = 0, x not equal to 0.

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