Complex Numbers And Quadratic Equations question 693
Question: If $ 0<a<b<c $ and the roots $ \alpha ,\beta $ of the equation $ ax^{2}+bx+c=0 $ are imaginary then incorrect statement is
Options:
A) $ |\alpha =|\beta | $
B) $ |\alpha |>1 $
C) $ |\beta |<1 $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Since the roots are imaginary
$ \therefore D < 0 $ and roots occur as conjugate pair, i.e. $ \beta = \bar{\alpha } $
$ \therefore | \beta |=| {\bar{\alpha }} |=| \alpha | $ Also, let $ \alpha =\frac{-b+i\sqrt{4ac-b^{2}}}{2a} $
$ \therefore | \alpha |=\sqrt{\frac{b^{2}}{4a^{2}}+\frac{4ac-b^{2}}{4a^{2}}}=\sqrt{\frac{c}{a}} $ $ | \alpha |>1( \because c>a ) $
$ \therefore | a |=| \beta |>1 $
BETA
