Complex Numbers And Quadratic Equations question 701
Question: If $ \alpha $ and $ \beta $ $ (\alpha <\beta ) $ are the roots of the equation $ x^{2}+bx+c=0, $ where, $ c<0<b, $ then
Options:
A) $ 0<\alpha <\beta $
B) $ \alpha <0<\beta <|\alpha | $
C) $ \alpha <\beta <0 $
D) $ \alpha <0<|\alpha |<\beta $
Show Answer
Answer:
Correct Answer: B
Solution:
Given $ \alpha < \beta , c < 0, b > 0, $
$ \therefore \alpha +\beta =-b<0 and \alpha \beta =c<0 $ Clearly, $ \alpha $ and $ \beta $ have opposite signs and $ \alpha < \beta $
$ \therefore \alpha < 0 and \beta >0\Rightarrow \alpha <0<\beta $ Further $ \alpha +\beta <0\Rightarrow \beta <-\alpha \Rightarrow | \beta |<| -\alpha | $
$ \Rightarrow \beta <|\alpha |(\beta >0\Rightarrow | \beta |=\beta ) $ Hence, $ \alpha < 0 < \beta < | \alpha | $
BETA
