Home
Class 12
MATHS
a ,b ,c are three complex numbers on the...

`a ,b ,c` are three complex numbers on the unit circle `|z|=1,` such that `a b c=a+b+cdot` Then find the value of `|a b+b c+c a|dot`

Text Solution

Verified by Experts

The correct Answer is:
1
a, b, c are complex numbers on the unit circle `|z| = 1 `
` therefore |a| = |b| = |c| = 1 `
` therefore a bara = bbarb = cbarc = 1 `
` " " abc = a + b + c `
` rArr |abc| = |a + b + c | `
` " "= |bara + bar b + bar c | `
` " " = |(1)/(a) + (1)/(b) + (1)/(c ) | `
` " " = | (ab + bc + ca )/( abc ) | `
` therefore | ab + bc + ca | = |abc | |abc | = (|a| |b| |c|)^(2) = 1 `
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.9|8 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.10|10 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.7|6 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

  • CONIC SECTIONS

    CENGAGE ENGLISH|Exercise All Questions|101 Videos

Similar Questions

Explore conceptually related problems

a ,b ,c are three complex numbers on the unit circle |z|=1, such that a b c=a+b+c dot Then find the value of |a b+b c+c a|dot

a , b , c are three complex numbers on the unit circle |z|=1 , such that abc=a+b+c . Then |ab+bc+ca| is equal to

If a+b+c=0 and a^2+b^2+c^2=16 , find the value of a b+b c+c a

If a b c=0 , then find the value of {(x^a)^b}^c (a)1 (b) a (c)b (d) c

Let a ,b and c be real numbers such that a+2b+c=4 . Find the maximum value of (a b+b c+c a)dot

If a,b and c are three consecutive positive integers such that 1/(a!)+1/(b!)=lambda/(c!) then find the value of root under lambda .

Let A,B,C be the three angles such that A+B+C=pi , tan A.tan B=2 , then find the value of (cos A cos B)/(cos C)

If a+b+c=9 and a b+b c+c a=23 , find the value of a^2+b^2+c^2

If vec a ,\ vec b ,\ vec c are unit vectors such that vec a+ vec b+ vec c= vec0 find the value of vec adot vec b+ vec bdot vec c+ vec cdot vec adot'

If a,b,c are positive real numbers such that a + b +c=18, find the maximum value of a^2b^3c^4

Home
Profile