Home
Class 12
MATHS
Let z(1),z(2) and z(3) be three complex ...

Let `z_(1),z_(2)` and `z_(3)` be three complex number such that `|z_(1)-1|= |z_(2) - 1| = |z_(3) -1| and arg ((z_(3) - z_(1))/(z_(2) -z_(1))) = (pi)/(6)`
then prove that `z_(2)^(3) + z_(3)^(3) + 1 = z_(2) + z_(3) + z_(2)z_(3)`.

Text Solution

Verified by Experts

Clearly, `z_(1),z_(2)` and `z_(3)` lie on a circle having center at `(1,0).`
Also, `angle BAC=30^(@)`

So, `DeltaBOC` is equilateral.
`impliesz_(2)^(2)+z_(3)^(3)+1=z_(2)+z_(3)+z_(2)z_(3)`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise SLOVED EXAMPLES|15 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.1|4 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise Comprehension|11 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

  • CONIC SECTIONS

    CENGAGE ENGLISH|Exercise All Questions|101 Videos

Similar Questions

Explore conceptually related problems

Let z_(1),z_(2) be two complex numbers such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| . Then,

If z_(1) and z_(2) are to complex numbers such that two |z_(1)|=|z_(2)|+|z_(1)-z_(2)| , then arg (z_(1))-"arg"(z_(2))

If z_(1) , z_(2) are complex numbers such that Re(z_(1))=|z_(1)-2| , Re(z_(2))=|z_(2)-2| and arg(z_(1)-z_(2))=pi//3 , then Im(z_(1)+z_(2))=

If z_(1) and z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 , then

Let z_(1) and z_(2) be any two non-zero complex numbers such that 3|z_(1)|=2|z_(2)|. "If "z=(3z_(1))/(2z_(2)) + (2z_(2))/(3z_(1)) , then

Let z_(1),z_(2),z_(3),……z_(n) be the complex numbers such that |z_(1)|= |z_(2)| = …..=|z_(n)| = 1 . If z = (sum_(k=1)^(n)z_(k)) (sum_(k=1)^(n)(1)/(z_(k))) then prove that : z is a real number .

If z_(1),z_(2) and z_(3) be unimodular complex numbers, then the maximum value of |z_(1)-z_(2)|^(2)+|z_(2)-z_(3)|^(2)+|z_(3)-z_(1)|^(2) , is

Let z_(1) and z_(2) be two given complex numbers such that z_(1)/z_(2) + z_(2)/z_(1)=1 and |z_(1)| =3 , " then " |z_(1)-z_(2)|^(2) is equal to

if z_(1),z_(2),z_(3),…..z_(n) are complex numbers such that |z_(1)|=|z_(2)| =….=|z_(n)| = |1/z_(1) +1/z_(2) + 1/z_(3) +….+1/z_(n)| =1 Then show that |z_(1) +z_(2) +z_(3) +……+z_(n)|=1

z_(1) "the"z_(2) "are two complex numbers such that" |z_(1)| = |z_(2)| . "and" arg (z_(1)) + arg (z_(2) = pi," then show that "z_(1) = - barz_(2).

CENGAGE ENGLISH-COMPLEX NUMBERS-ILLUSTRATION
  1. If z1, z2, z3 are three complex numbers such that 5z1-13 z2+8z3=0, the...

    Text Solution

    |

  2. If z=z0+A( z -( z )0), w h e r eA is a constant, then prove that loc...

    Text Solution

    |

  3. z1a n dz2 are the roots of 3z^2+3z+b=0. if O(0),(z1),(z2) form an equi...

    Text Solution

    |

  4. Let z(1),z(2) and z(3) be three complex number such that |z(1)-1|= |z...

    Text Solution

    |

  5. Let the complex numbers z(1),z(2) and z(3) be the vertices of an equ...

    Text Solution

    |

  6. In the Argands plane what is the locus of z(!=1) such that a rg{3/2((2...

    Text Solution

    |

  7. If ((3-z1)/(2-z1))((2-z2)/(3-z2))=k(k >0) , then prove that points A(z...

    Text Solution

    |

  8. If z1, z2, z3 are complex numbers such that (2//z1)=(1//z2)+(1//z3), t...

    Text Solution

    |

  9. A(z1),B(z2),C(z3) are the vertices of he triangle A B C (in anticlockw...

    Text Solution

    |

  10. If one of the vertices of the square circumscribing the circle abs(z-1...

    Text Solution

    |

  11. Let z1=10+6i and z2=4+6idot If z is any complex number such that the a...

    Text Solution

    |

  12. Complex numbers z(1),z(2)andz(3) are the vertices A,B,C respectivelt o...

    Text Solution

    |

  13. Let z1, z2a n dz3 represent the vertices A ,B ,a n dC of the triangle ...

    Text Solution

    |

  14. F a=cos(2pi//7)+isin(2pi//7) , then find the quadratic equation whose ...

    Text Solution

    |

  15. If omega is an imaginary fifth root of unity, then find the value of l...

    Text Solution

    |

  16. If 1, alpha(1), alpha(2), alpha(3),…….,alpha(s) are ninth roots of uni...

    Text Solution

    |

  17. find the sum of squares of all roots of the equation. x^(8) - x^(7) ...

    Text Solution

    |

  18. Find roots of the equation (z + 1)^(5) = (z - 1)^(5).

    Text Solution

    |

  19. If the roots of (z-1)^n=i(z+1)^n are plotted in ten Arg and plane, the...

    Text Solution

    |

  20. Let 1, z(1),z(2),z(3),…., z(n-1) be the nth roots of unity. Then pro...

    Text Solution

    |