Q. 44.3( 7 Votes )

# If z_{1} and z_{2} are two complex number such that |z_{1}| = |z_{2}| and arg (z_{1}) + arg (z_{2}) = π, then show that

Answer :

Given:

⇒ |z_{1}|=|z_{2}| and arg(z_{1})+arg(z_{2})=

Let us assume arg(z_{1})=θ

⇒ arg(z_{2})=-θ

We know that z=|z|(cosθ+isinθ)

⇒ z_{1}=|z_{1}|(cosθ+isinθ)-----------------(1)

⇒ z_{2}=|z_{2}|(cos(-θ)+isin(-θ))

⇒ z_{2}=|z_{2}|(-cosθ+isinθ)

⇒ z_{2}=-|z_{2}|(cosθ-isinθ)

Now we find the conjugate of z_{2}

⇒ =-|z_{2}|(cosθ+isinθ) (∵ )

Now,

⇒

⇒ (∵ |z_{1}|=|z_{2}|)

⇒ z_{1}=-

∴ Thus proved.

Rate this question :

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 1 – i

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form: –i

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form:

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form:

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form:

RS Aggarwal - MathematicsFind the modulus of each of the following complex numbers and hence express each of them in polar form: –1 + i

RS Aggarwal - MathematicsWrite 2i in polar form.

RS Aggarwal - MathematicsFind the principal argument of (–2i).

RS Aggarwal - Mathematics