If z ≠ 0 be a complex number such that |z-1/z|=2, then the maximum value of |z| is :

Solution

Maths sir · 4 steps

Use the triangle inequality on z - 1/z: it gives a quadratic inequality in r = |z|.

1. Triangle inequality. For any two complex numbers, |a - b| ≥ |a| - |b|. Take a = z and b = 1/z, and write r = |z|, so |1/z| = 1/r.

   r - 1/r ≤ |z - 1/z| = 2

2. Solve for r. Multiply by r > 0 to get r² - 2r - 1 ≤ 0. The roots of r² - 2r - 1 = 0 are 1 ± √2, so r lies between them, and r ≤ √2 + 1.
3. Check it is reached. Take z = √2 + 1. Then 1/z = √2 - 1, and z - 1/z = 2, so the condition holds exactly. The bound is attained.
4. So. The maximum value of |z| is √2 + 1, option (d).

The idea to keep. When a condition fixes |z ± 1/z|, apply the triangle inequality with r = |z|. You get a quadratic in r, and its roots give the maximum and minimum of |z|.

Answer from the NTA final answer key. Solution by PrepWiser.