If z1 and z2 are two complex numbers such that |z1|

Question

If z1 and z2 are two complex numbers such that |z1|<1<|z2| then |(1-z1 barz2/z1-z2)| (A) equal to 1 (B) greater than 1 (C) less than 1 (D) none of these

Tardigrade Admin · Accepted Answer

Given that |z1|<1<|z2| Let |(1-z1 barz2/z1-z2)|<1 ⇒|1-z1 barz2|<|z1-z2| ⇒|1-z1 barz2|2<|z1-z2|2 ⇒ (1-z1 barz2)( overline1-z1 barz2)<(z1-z2)( overlinez1-z2) ⇒ (1-z1 barz2)(1- barz1 z2)<(z1-z2)( barz1- barz2) ⇒ 1-z1 barz2- barz1 z2+z1 barz1 z2 barz2< z1 barz1-z1 barz2- barz1 z2+z2 barz2 ⇒ 1+|z1|2|z2|2< |z1|2+|z2|2 ⇒ (1-|z1|2)(1-|z2|2)<0 which is obviously true as |z1|<1<|z2| and hence, |z1|2<1 <|z2|2.

Q. If z1​ and z2​ are two complex numbers such that ∣z1​∣<1<∣z2​∣ then ∣∣​z1​−z2​1−z1​zˉ2​​∣∣​

equal to 1

greater than 1

less than 1

none of these

Q. If $z_{1}$ and $z_{2}$ are two complex numbers such that $∣ z_{1} ∣ < 1 < ∣ z_{2} ∣$ then $∣ ∣ \frac{1 - z _{1} z ˉ _{2}}{z _{1} - z _{2}} ∣ ∣$