First slide

Algebra of complex numbers

Question

If $ω \neq 1$ is a cube root of unity and satisfies
$\frac{1}{a + ω} + \frac{1}{b + ω} + \frac{1}{c + ω} = 2 ω^{2}$
and $\frac{1}{a + ω^{2}} + \frac{1}{b + ω^{2}} + \frac{1}{c + ω^{2}} = 2 ω,$
then the value of $\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c + 1}$ then

Moderate

A

2
B

-2
C

$= 1 + ω^{2}$
D

none of these

Solution

Note that $ω, ω^{2}$ are roots of $\frac{1}{a + x} + \frac{1}{b + x} + \frac{1}{c + x} = \frac{2}{x}$
$\begin{array}{l} \Leftrightarrow x [b c + c a + a b + 2 (a + b + c) x + 3 x^{2}] \\ = 2 [a b c + (b c + c a + a b) x + (a + b + c) x^{2} + x^{3}] \end{array}$
If $α$ is the third root of this equation, then
$α + ω + ω^{2} = 0 \Rightarrow α = 1$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App