Let z1, Z2 and z3 be three complex numbers satisfying |z| = 1 and 4z3 = 3(z1 + z2), then |z1 – z2| is equal to

Let z₁, Z₂ and z₃ be three complex numbers satisfying |z| = 1 and 4z₃= 3(z₁ + z₂), then |z₁ - z₂| is equal to

A
$\frac{2}{3}$
B
$\frac{\sqrt{5}}{3}$
C
$\frac{3}{2}$
D
$\frac{2 \sqrt{5}}{3}$

Solution:

Given that 4z₃ = 3(z₁ + z₂)

$∴ \frac{z_{1} + z_{2}}{2} = \frac{2}{3} z_{3}$

Therefore, mid-point of AB is the point which divides OM in the ratio 2 : 1.

or $OM = \frac{2}{3} OC = \frac{2}{3} |z_{3}| = \frac{2}{3}$

$∴ |z_{1} - z_{2}| = AB = 2 AM = 2 \sqrt{1 - \frac{4}{9}} = \frac{2 \sqrt{5}}{3}$

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