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  • Can someone help me with this, - Complex numbers and quadratic equations - JEE Main-8

Let z,\omega  be complex numbers such that  \overline{z}+i\overline{\omega}=0  and  argz\omega =\pi Then arg z equals

  • Option 1)

    3\pi/4

  • Option 2)

    \pi/2

  • Option 3)

    \pi/4

  • Option 4)

    5\pi/4

 

Answers (2)

best_answer

As we learnt in

Property of Argument of a Complex Number -

Arg(z.w)=Arg(z)+Arg(w)+2n\pi

- wherein

n\epsilon Integer and it is chosen such that Arg(z.w) lies in the principal value range of Argument.  

 

 \overline{z}+i\overline{\omega}=0

\Rightarrow\ \; |z|=|\overline{\omega}|

|\overline{z}|=|-i\overline{\omega}|

arg|z\omega|=argz+arg\omega=\pi

 

\\\therefore\ \; argz+arg\omega=\pi \\ arg(i\omega)+arg\omega=arg(i)+arg\omega+arg\omega=\frac{\pi}{2} +2arg\omega=\pi \\2arg\omega=\pi-\frac{\pi}{2}=\frac{\pi}{2} \\arg\omega=\frac{\pi}{4} \\argz+\frac{\pi}{4}=\pi \\argz=\frac{3\pi}{4}\\\therefore\ \; argz+arg\omega=\pi \\ arg(i\omega)+arg\omega=arg(i)+arg\omega+arg\omega=\frac{\pi}{2} +2arg\omega=\pi \\2arg\omega=\pi-\frac{\pi}{2}=\frac{\pi}{2} \\arg\omega=\frac{\pi}{4} \\argz+\frac{\pi}{4}=\pi \\argz=\frac{3\pi}{4}

 

Correct option is 1.

 


Option 1)

3\pi/4

Option 2)

\pi/2

Option 3)

\pi/4

Option 4)

5\pi/4

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