Let z1, z2 be two non-zero complex numbers such thatz1−z22+z1z2=z¯1z2+z1z¯2then z2z1 lies in the interval

Write the given equation as z12+z22−2z1z2+z1z2=z¯1z2+z1z¯2⇒z1−z22=z1z2⇒z−1z=1 where z=z2/z1If |z|≥1, we get |z|−1|z|≤z−1z=1⇒|z|2−|z|−1≤0⇒|z|−122≤54⇒1≤|z|≤5+12⇒1≤z2z1≤3+52If |z|<1, 1<z1z2≤3+52⇒3−52≤z2z1<1Thus, 12(3−5)≤z2z1≤12(3+5)

Let z1, z2 be two non-zero complex numbers such thatz1−z22+z1z2=z¯1z2+z1z¯2then z2z1 lies in the interval

Write the given equation as z12+z22−2z1z2+z1z2=z¯1z2+z1z¯2⇒z1−z22=z1z2⇒z−1z=1 where z=z2/z1If |z|≥1, we get |z|−1|z|≤z−1z=1⇒|z|2−|z|−1≤0⇒|z|−122≤54⇒1≤|z|≤5+12⇒1≤z2z1≤3+52If |z|<1, 1<z1z2≤3+52⇒3−52≤z2z1<1Thus, 12(3−5)≤z2z1≤12(3+5)

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Let $z_{1}, z_{2}$ be two non-zero complex numbers such that
${(|z_{1}| - |z_{2}|)}^{2} + |z_{1} z_{2}| = {\bar{z}}_{1} z_{2} + z_{1} {\bar{z}}_{2}$
then $|\frac{z_{2}}{z_{1}}|$ lies in the interval

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$[\frac{1}{2} (3 - \sqrt{5}), \frac{1}{2} (3 + \sqrt{5})]$

$[\frac{1}{2} (\sqrt{5} - 1), \frac{1}{2} (\sqrt{5} + 1)]$

$[1, \frac{1}{2} (3 + \sqrt{5})]$

$[\frac{1}{2} (3 - \sqrt{5}), 1]$

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Write the given equation as

$\begin{array}{l} {|z_{1}|}^{2} + {|z_{2}|}^{2} - 2 |z_{1}| |z_{2}| + |z_{1}| |z_{2}| \\ = {\bar{z}}_{1} z_{2} + z_{1} {\bar{z}}_{2} \\ \Rightarrow {|z_{1} - z_{2}|}^{2} = |z_{1} z_{2}| \end{array}$

$\Rightarrow |z - \frac{1}{z}| = 1$ where $z = \sqrt{z_{2} / z_{1}}$

If $| z | \geq 1$ , we get $| z | - \frac{1}{| z |} \leq |z - \frac{1}{z}| = 1$

$\begin{array}{l} \Rightarrow | z |^{2} - | z | - 1 \leq 0 \\ \Rightarrow {(| z | - \frac{1}{2})}^{2} \leq \frac{5}{4} \Rightarrow 1 \leq | z | \leq \frac{\sqrt{5} + 1}{2} \\ \Rightarrow 1 \leq |\frac{z_{2}}{z_{1}}| \leq \frac{3 + \sqrt{5}}{2} \end{array}$