Question

if |z| = 1 show that z-1/z+1,(z ≠ -1) is a pure imaginary number. what will you conclude if z = 1 ?​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

Answer

Let z=x+iy. Then ∣z∣

2

=x

2

+y

2

.

Therefore the condition ∣z∣=1 is equivalent to

x

2

+y

2

=1.

Now

z+1

z−1

=

x+iy+1

x+iy−1

=

(x+1+iy)(x+1−iy)

(x−1+iy)(x+1−iy)

=

(x+1)

2

+y

2

(x

2

+y

2

−1)+2iy

=

(x+1)

2

+y

2

2iy

by (1)

Hence

z+1

z−1

is purely imaginary when ∣z∣=1

provided z



=−1.

When z=1, we have

z+1

z−1

=0.

Now recall that according to the definition 2 given in $$\S2$$, 0 is a pure imaginary number, since the point 0 which corresponds to z=0 lies on both real and imaginary axes.

So in this case also,

z+1

z−1

is a pure imaginary number.

Step-by-step explanation:

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