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Mathematics Question on complex numbers

If z1 and z2 are two complex numbers satisfying the equation∣z1+z2z1−z2∣=1|\frac{z_1+z_2}{z_1-z_2}|=1, then z1z2\frac{z_1}{z_2} may be

A

real positive

B

real negative

C

zero

D

purely imaginary

Answer

zero

Explanation

Solution

The correct answer is/are option(s):
(C): zero
(D): purely imaginary

∣z1+z2z1−z2∣=1|\frac{z_1+z_2}{z_1-z_2}|=1

Now,

⇒∣z1+z2∣=∣z1−z2∣\Rightarrow |z_1+z_2|=|z_1-z_2|

⇒∣z1+z2∣2=∣z1−z2∣2\Rightarrow |z_1+z_2|^2=|z_1-z_2|^2

⇒(z1+z2)(z1+z2)=(z1−z2)(z1−z2)\Rightarrow (z_1+z_2)(z_1+z_2)=(z_1-z_2)(z_1-z_2)

⇒∣z1∣2+∣z2∣2+z1z2+z1z2=∣z1∣2+∣z2∣2−z1z2−z1z2\Rightarrow|z_1|^2+|z_2|^2+z_1z_2+z_1z_2=|z_1|^2+|z_2|^2-z_1z_2-z_1z_2

⇒2(z1z2+z2z1)=0\Rightarrow 2(z_1z_2+z_2z_1)=0

⇒z1z2+z1z2=0\Rightarrow \frac{z_1}{z_2}+\frac{z_1}{z_2}=0

⇒2Re(z1z2)=0=z1z2\Rightarrow 2 Re (\frac{z_1}{z_2})=0=\frac{z_1}{z_2}

Hence it is purely imazinary.