Home
Class 12
MATHS
If P(z(1)),Q(z(2)),R(z(3)) " and " S(z(4...

If `P(z_(1)),Q(z_(2)),R(z_(3)) " and " S(z_(4))` are four complex numbers representing the vertices of a rhombus taken in order on the complex plane, which one of the following is held good?

A

`(z_(1)-z_(4))/(z_(2)-z_(3))` is purely real

B

`amp (z_(1) -z_(4))/(z_(2) - z_(4)) = amp (z_(2)-z_(4))/(z_(3) - z_(4))`

C

`(z_(1) - z_(3))/(z_(2)-z_(4))` is pureluy imaginary

D

is not necessary that `|z_(1)-z_(3)|ne |z_(2) - z_(4)|`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D

(1) `PS||QR rArr arg((z_(1)-z_(4))/(z_(2) - z_(3))) = 0 rArr (z_(1) - z_(4))/(z_(2) -z_(4))` is purely real
(2) since diaginal bisect the angle
`rArr amp ((z_(1)-z_(4))/(z_(2)-z_(4)))= amp ((z_(2)-z_(4))/(z_(3) -z_(4)))`
(3) Digonals of rohombus are prependicular. Henc,`(z_(1)-z_(3))//(z_(2)-z_(4))` is purley imagainary .
(4) Digonal of rhombus are not necessarily equal.
Hence, `|z_(1) -z_(3)| ne |z_(2) -z_(4)|` not necessarily .
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise LINKED COMPREHENSION TYPE|36 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise NUMERICAL VALUE TYPES|33 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise single correct Answer type|92 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

  • CONIC SECTIONS

    CENGAGE ENGLISH|Exercise All Questions|101 Videos

Similar Questions

Explore conceptually related problems

If z_(1), z_(2), z_(3) are three complex numbers representing three vertices of a triangle, then centroid of the triangle be (z_(1) + z_(2) + z_(3))/(3)

If z_(1) , z_(2), z_(3) are three complex numbers in A.P., then they lie on :

If the points z_(1),z_(2),z_(3) are the vertices of an equilateral triangle in the Argand plane, then which one of the following is not correct?

If z_(1),z_(2),z_(3) and u,v,w are complex numbers represending the vertices of two triangles such that z_(3)=(1-t)z_(1)+tz_(2) and w = (1 -u) u +tv , where t is a complex number, then the two triangles

All complex numbers 'z' which satisfy the relation |z-|z+1||=|z+|z-1|| on the complex plane lie on the

Let p and q are complex numbers such that |p|+|q| lt 1 . If z_(1) and z_(2) are the roots of the z^(2)+pz+q=0 , then which one of the following is correct ?

If |(z-z_(1))//(z-z_(2))| = 3 , where z_(1) and z_(2) are fixed complex numbers and z is a variable complex complex number, then z lies on a

For complex number z,|z-1|+|z+1|=2, then z lies on

If z_(1), z_(2), z_(3), z_(4) are complex numbers, show that they are vertices of a parallelogram In the Argand diagram if and only if z_(1) + z_(3)= z_(2) + z_(4)

Let z_1, z_2 be two complex numbers represented by points on the circle |z_1|= and and |z_2|=2 are then

CENGAGE ENGLISH-COMPLEX NUMBERS-MULTIPLE CORRECT ANSWERS TYPE
  1. A rectangle of maximum area is inscribed in the circle |z-3-4i|=1. If ...

    Text Solution

    |

  2. If |z1|=15 and |z2-3-4i|=5,t h e n

    Text Solution

    |

  3. If P(z(1)),Q(z(2)),R(z(3)) " and " S(z(4)) are four complex numbers re...

    Text Solution

    |

  4. about to only mathematics

    Text Solution

    |

  5. If a complex number z satisfies |z| = 1 and arg(z-1) = (2pi)/(3), then...

    Text Solution

    |

  6. If |z-1|=1, then

    Text Solution

    |

  7. If z(1) = 5 + 12i and |z(2)| = 4, then

    Text Solution

    |

  8. Let z(1),z(2),z(3) be the three nonzero comple numbers such that z(1)...

    Text Solution

    |

  9. z(1) " and " z(2) are the roots of the equation z^(2)-az+b=0, " where ...

    Text Solution

    |

  10. If |(z-z(1))//(z-z(2))| = 3, where z(1) and z(2) are fixed complex ...

    Text Solution

    |

  11. If z=x+i y , then the equation |(2z-i)/(z+1)|=m does not represents a ...

    Text Solution

    |

  12. System of equaitons |z+3|-|z-3| = 6 and |z-4|=r where r in R^(+) has

    Text Solution

    |

  13. Let the equation of a ray be |z-2|-|z-1-i| = sqrt(2). If it strikes t...

    Text Solution

    |

  14. Given that the two curves a r g(z)=pi/6 and |z-2sqrt(3)i|=r intersect ...

    Text Solution

    |

  15. On the Argand plane ,let z(1) = - 2+ 3z,z(2)= - 2-3z and |z| = 1. T...

    Text Solution

    |

  16. Let S = { z: x = x+ iy, y ge 0,|z-z(0)| le 1}, where |z(0)|= |z(0) - o...

    Text Solution

    |

  17. If P and Q are represented by the complex numbers z(1) and z(2) such ...

    Text Solution

    |

  18. Locus of complex number satisfying "a r g"[(z-5+4i)//(z+3-2i)]=pi//4 i...

    Text Solution

    |

  19. Equation of tangent drawn to circle abs(z)=r at the point A(z(0)), is

    Text Solution

    |

  20. If n is a natural number gt 2, such that z^(n) = (z+1)^(n), then (a) ...

    Text Solution

    |