If z is a complex number such that \left | z \right |\geq 2,   then the minimum value of \left | z+\frac{1}{2} \right |:  

  • Option 1)

    is strictly greater than \frac{5}{2}

  • Option 2)

    is strictly greater than \frac{3}{2}  but less than \frac{5}{2}

  • Option 3)

    is equal to  \frac{5}{2}

  • Option 4)

    lies in the interval (1, 2)

 

Answers (2)
N neha
H Himanshu

As we have learned

Triangle Law of Inequality in Complex Numbers -

|z_{1}-z_{2}|\geq \left || z_{1} \right |-| z_{2} \right |||

- wherein

|.| denotes modulus of z in complex numbers

 

 \left | z+\frac{1}{z} \right |= \left | z-(-\frac{1}{z}) \right |

\geq \left | |z|- (-\frac{1}{z}) \right |

= |z| - 1/z (\because |z|\geq 2)

 

\geq 2-1/2 = 3/2

 

\left | z+1/z \right |\geq 3/2

3/2 lies in the interval (1,2)

 

 

 

 

 


Option 1)

is strictly greater than \frac{5}{2}

Option 2)

is strictly greater than \frac{3}{2}  but less than \frac{5}{2}

Option 3)

is equal to  \frac{5}{2}

Option 4)

lies in the interval (1, 2)

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