Question

If `|z - 4/z|` = 2, then the maximum value of |z| is equal to ______.

विकल्प

• `sqrt(5) + 1`

• 2

• `2 + sqrt(2)`

• `sqrt(3) + 1`

Answer 1

If `|z - 4/z|` = 2, then the maximum value of |z| is equal to `underlinebb(sqrt(5) + 1)`.

Explanation:

Given that `|z - 4/z|` = 2

|z| = `|z - 4/z + 4/z| ≤ |z - 4/z| + 4/|z|`

⇒ `|z| ≤ 2 + 4/|z|`

⇒ |z|² – 2|z| – 4 ≤ 0

⇒ `(|z| - (2 + sqrt(20))/2)(|z| - (2 - sqrt(20))/2) ≤ 0`

⇒ `(|z| - (1 + sqrt(5)))(|z| - (1 - sqrt(5))) ≤ 0`

⇒ `(-sqrt(5) + 1) ≤ |z| ≤ (sqrt(5) + 1)`

⇒ |z|_max = `sqrt(5) + 1`