Home
Class 11
MATHS
If z1,z2, z3 are complex numbers such th...

If `z_1,z_2, z_3` are complex numbers such that `|z_1|=|z_2|=|z_3|=|1/z_1+1/z_2+1/z_3|=1` then `|z_1+z_2+z_3|` is equal to

Text Solution

Verified by Experts

We have given `z_1,z_2,z_3`as complex numbers such that
`∣(z_1)∣=∣z_2∣=∣z_3​∣=|(1/z_1)+(1/z_2)+(1/z_3)|=1`
`z,bar(z_1)=1 implies bar(z_1) =1/z_1` and so on.
And also `∣z_1∣=∣bar(z_1)∣`, i.e., `bar(z_1)` is the conjugate of`z`,
`|z_1+z_2+z_3|= bar(|z_1+z_2+z_3|)`
`|bar(z_1)+bar(z_2)+bar(z_3)|` `=|(1/z_1)+(1/z_2)+(1/z_3)|`
Therefore `|z_1+z_2+z_3|=1`
Promotional Banner

Topper's Solved these Questions

  • COMBINATIONS

    RD SHARMA|Exercise Solved Examples And Exercises|117 Videos

  • DERIVATIVES

    RD SHARMA|Exercise Solved Examples And Exercises|236 Videos

Similar Questions

Explore conceptually related problems

If z_1,z_2 and z_3 are complex numbers such that |z_1|=|z_2|=|z_3|= |1/z_1+1/z_2+1/z_3|=1, then |z_1+z_2+z_3| is (A) equal to 1 (B) gt1 (C) gt3 (D) equal to 3

If z_1,z_2,z_3 are complex numbers such that |z_1|=z_2|=|z_3|=|z_1+z_2+z_3|=1, then (1/z_1+1/z_2+1/z_3| is (A) equal to 1 (B) les than (C) greater than 3 (D) equal to 3

If z_(1),z_(2),z_(3) are complex numbers such that |z_(1)|=|z_(2)|=|z_(3)|=|(1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3))|=1 then |z_(1)+z_(2)+z_(3)| is equal to

If z_(1),z_(2),z_(3) are complex numbers such that |z_(1)|=|z_(2)|=|z_(3)|=1|(1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3))|=1 Then find the value of |z_(1)+z_(2)+z_(3)| is :

If z_(1),z_(2),z_(3) are complex numbers such that |z_(1)|=|z_(2)|=|z_(3)|=|(1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3))|=1 Then find the value of |z_(1)+z_(2)+z_(3)| is :

Let z_1,z_2,z_3 be three complex numbers such that |z_1|=1,|z_2|=2,|z_3|=3 and |z_1+z_2+z_3|=1. Find |9z_1z_2+4z_1z_3+z_2z_3| .

RD SHARMA-COMPLEX NUMBERS-Solved Examples And Exercises
  1. Write the least positive integral value of n for which ((1+i)/(1-i))^n...

    Text Solution

    |

  2. If f(z)=(7-z)/(1-z^2) , where z=1+2i , then |f(z)| is (a)(|z|)/2 ...

    Text Solution

    |

  3. If z1,z2, z3 are complex numbers such that |z1|=|z2|=|z3|=|1/z1+1/z2+1...

    Text Solution

    |

  4. What is the locus of z , if amplitude of (z-2-3i) is pi/4?

    Text Solution

    |

  5. Write the sum of the series i+i^2+i^3+....... upto 1000 terms.

    Text Solution

    |

  6. The sum and product of two complex numbers are real if and only if the...

    Text Solution

    |

  7. If ((a+i)^2)/((2a-i))=p+i q , show that: p^2+q^2=((a^2+1)^2)/((4a^2+1)...

    Text Solution

    |

  8. Write the value of a r g(z)+a r g(barz ) .

    Text Solution

    |

  9. Find the conjugate of (1-i)/(1+i)

    Text Solution

    |

  10. Find the conjugate of 1/(3+4i)dot

    Text Solution

    |

  11. If (a+i b)/(c+i d)=x+i y , prove that (a-i b)/(c-i d)=x-i ya n d(a^2+b...

    Text Solution

    |

  12. Find real values of xa n dy for which the complex numbers -3+i x^2ya n...

    Text Solution

    |

  13. Find the real values of x and y for which (1+i)y^2+(6+i)=(2+i)x

    Text Solution

    |

  14. Find the real values of xa n dy , if (3x-7)+2i y=-5y+(5+x)i

    Text Solution

    |

  15. If (x+i y)^(1//3)=a+i b ,x ,y ,a b in Rdot Show that x/a+y/b=4(a^2-b^...

    Text Solution

    |

  16. If a+i b=(c+i)/(c-i) , where c is real, then :a^2+b^2

    Text Solution

    |

  17. Evaluate the following: i^(135)

    Text Solution

    |

  18. Evaluate the following: i^(19)

    Text Solution

    |

  19. Evaluate the following: i^(-999)

    Text Solution

    |

  20. Evaluate the following: \ (-sqrt(-1))^(4n+3), ninNN

    Text Solution

    |