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Question: If z be any complex number (z ¹ 0), then arg \(\left( \frac{z - i}{z + i} \right)\) = \(\frac{\pi}{2...

If z be any complex number (z ¹ 0), then arg (z−iz+i)\left( \frac{z - i}{z + i} \right) = π2\frac{\pi}{2} represents the curve –

A

|z| = I

B

|z| = 1; Re (z) > 0

C

|z| = 1 ; Re (z) < 0

D

None of these

Answer

|z| = 1 ; Re (z) < 0

Explanation

Solution

Sol. arg Z = π2\frac{\pi}{2}

Ž Z = P.I. = ri (r + ive)

\ z−iz+i\frac{z - i}{z + i} = ir

L.H.S. = (z−i)(z+i‾)(z+i)(z+i‾)\frac{(z - i)(\overline{z + i})}{(z + i)(\overline{z + i})}

= (z−i)(zˉ+iˉ)∣z+i∣2\frac{(z - i)(\bar{z} + \bar{i})}{|z + i|^{2}}

= (z−i)(zˉ−i)+iveR\frac{(z - i)(\bar{z} - i)}{+ iveR}=zzˉ+i2−i(z+zˉ)+iveR\frac{z\bar{z} + i^{2} - i(z + \bar{z})}{+ iveR}

= (x2+y2−1)−i.(2x)+iveR\frac{(x^{2} + y^{2} - 1) - i.(2x)}{+ iveR} = ir

\ Real part = 0 Ž x2 + y2 = 1 or |z| = 1

Imaginary part – 2x = R . r = + ive

\ x < 0 i.e., Re (z) < 0

Both imply (3).