KVPY SX SB Exam 2016 Question Paper

Question : 90 of 120

Marks: +1, -0

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

Solution:

| a | < 1 and

w=({z-a}/{1-a↖{-}z})

⇒ w-\ov azw=z-a

⇒ w + a = z(1 +\ov a w)

z={w+a}/{1+ a↖{-}w}

Given | z | < 1

|{w+a}/{1+a↖{-}w}| < 1

⇒{ | w + a |}^2 <{ | 1 + \ov a w |}^2

⇒ (w + a) (\ov w +\ov a ) < (1 +\ov a w) ( 1 + a \ov w )

⇒w\ov w + w\ov a + a\ov w + a\ov a

< 1 +\ov a w + a\ov w + a\ov a w\ov w

⇒ a \ov a . w \ov w - w\ov w - a\ov a + 1 > 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

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