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

Correct answer: (D) √2+1. Given z-1/z=2 We know that z-1/z≤z-1/z≤z+1/z Let z=r i.e. r-1/r≤2 and r+1/r≥2 which is always true. Now r-1/r≤2 ⇒r-1/r≥-2 & r-1/r≤2 ⇒r^2-1≤2r ⇒r-1^2≤2 ⇒r-1≤√2 ∴ z_max=1+√2

Q31 / 90 · JEE Main 2022 (29 Jul Shift 2) · Mathematics

If $z \neq 0$ be a complex number such that $|z - \frac{1}{z}| = 2$ , then the maximum value of $|z|$ is

A
$\sqrt{2}$
B
$1$
C
$\sqrt{2} - 1$
Correct answer
D
$\sqrt{2} + 1$

Solution

Given $|z - \frac{1}{z}| = 2$
We know that $||z| - \frac{1}{|z|}| \leq |z - \frac{1}{z}| \leq |z| + \frac{1}{|z|}$ Let $|z| = r$
i.e. $|r - \frac{1}{r}| \leq 2$ and
$r + \frac{1}{r} \geq 2$ which is always true.
Now $|r - \frac{1}{r}| \leq 2$
$\Rightarrow r - \frac{1}{r} \geq - 2 & r - \frac{1}{r} \leq 2$
$\Rightarrow r^{2} - 1 \leq 2 r$
$\Rightarrow {(r - 1)}^{2} \leq 2$
$\Rightarrow r - 1 \leq \sqrt{2}$
$∴ {|z|}_{\max} = 1 + \sqrt{2}$

Hence, the correct answer is option: 'D'.

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

Solution

Relevant Questions

If $z \neq 0$ be a complex number such that $|z - \frac{1}{z}| = 2$ , then the maximum value of $|z|$ is