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  • Answer please! - Let be any two non-zero complex numbers such that . If then : - Complex numbers and quadratic equations - JEE Main

Let z_{1} and z_{2} be any two non-zero complex numbers such that

 3|z_{1}| = 4|z_{2}|. If z= \frac{3z_{1}}{2z_{2}}+\frac{2z_{2}}{3z_{1}} then :

  • Option 1)

     

    Re(z)=0

  • Option 2)

     

    |z|=\sqrt{5/2}

  • Option 3)

    |z|=1/2{\sqrt{17/2}}

  • Option 4)

     

    Im(z) = 0

Answers (1)

best_answer

 

Property of Modulus of z(Complex Number) -

\left |\frac{z_{1}}{z_{2}} \right |=\frac{\left |z_{1} \right |}{\left |z_{2} \right |}

- wherein

|.| denotes modulus of complex number

 

  

 

Polar Form of a Complex Number -

z=r(cos\theta+isin\theta)

- wherein

r= modulus of z and \theta is the argument of z

 

Given , 3\left | Z_{1} \right | = 4\left | Z_{2} \right | \\

\Rightarrow \frac{\left | Z_{1} \right |}{\left | Z_{2} \right |} = \frac{4}{3}

\Rightarrow \frac{\left | 3Z_{1} \right |}{\left | 2Z_{2} \right |} = \frac{3}{2} \times \frac{4}{3} = 2

let \frac{\left | 3Z_{1} \right |}{\left | 2Z_{2} \right |} = a =2cos \theta + 2isin\theta

then \frac{\left | 2Z_{2} \right |}{\left | 3Z_{1} \right |} = \frac{1}{x} =\frac{1}{2}cos \theta -\frac{1}{2}isin\theta

z = a + \frac{1}{a} \\ = \frac{5}{2} cos \theta + \frac{3}{2} isin \theta

 


Option 1)

 

Re(z)=0

Option 2)

 

|z|=\sqrt{5/2}

Option 3)

|z|=1/2{\sqrt{17/2}}

Option 4)

 

Im(z) = 0

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