# If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = π2, then show that z¯w = –i. - Mathematics

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If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = pi/2, then show that barzw = –i.

#### Solution

Let z = r1 (cosθ1 + isinθ1) and w = r2 (cosθ2 + isinθ2)

zw = r1r2 [(cosθ1 + isinθ1)] [(cosθ2 + isinθ2)]

|zw| = r1r2 = 1

Now arg(z) – arg(w) = pi/2

θ1 – θ2 = pi/2

⇒ arg (z/w) = pi/2

barzw = r1 (cosθ1 – isinθ1) r2 (cosθ2 + isinθ2)

= r1 r2 [cosθ1 cosθ2 + icosθ1 sinθ2 – isinθ1 cosθ2 – i2 sinθ1 sinθ2]

= r1 r2 [(cosθ1 cosθ2 + sinθ1 sinθ2) + i(cosθ1 sinθ2 – sinθ1 cosθ2)]

= r1 r2 [cos(θ2 – θ1) + isin(θ2 – θ1)]

= r_1r_2 [cos((-pi)/2) + i sin((-pi)/2)]

= r_1r_2 [cos  pi/2 - i sin  pi/2]

= 1  .....[0 – i]

Here barzw = –i.

Hence proved.

Concept: Argand Plane and Polar Representation
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 24 | Page 92

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