KVPY2016MathematicsComplex Number
Let a be a fixed non-zero complex number with | a | < 1 and w = ( z - a 1- a z ) . Where z is a complex number. Then
Options
- Athere exists a complex number z with | z | 1
- B| w |>1 for all z such that | z | < 1
- C| w | < 1 for all z such that | z | < 1
- Dthere exists z such with | z | < 1 and | w |=1
Correct answer
C. | w | < 1 for all z such that | z | < 1
Step-by-step solution
| a | 0 | a |²| w |²-| w |²| a |²+1>0 (| a |²-1 ) (| w |²-1 )>0 Given | a | < 1 | w |²-1 < 0| w | < 1 &| z | < 1