Complex Numbers And Quadratic Equations question 519
Question: If the roots of the equation $ ax^{2}+bx+c=0 $ be $ \alpha $ and $ \beta $ , then the roots of the equation $ cx^{2}+bx+a=0 $ are [MNR 1988; RPET 2003]
Options:
A) $ -\alpha ,-\beta $
B) $ \alpha ,\frac{1}{\beta } $
C) $ \frac{1}{\alpha },\frac{1}{\beta } $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
$ \alpha ,\beta $ are roots of $ ax^{2}+bx+c=0 $
Þ $ \alpha +\beta =-\frac{b}{a} $ and $ \alpha \beta =\frac{c}{a} $ Let the roots of $ cx^{2}+bx+a=0 $ be $ {\alpha }’,{\beta }’ $ , then $ {\alpha }’+{\beta }’=-\frac{b}{c} $ and $ {\alpha }’{\beta }’=\frac{a}{c} $ but $ \frac{\alpha +\beta }{\alpha \beta }=\frac{-b/a}{c/a}=\frac{-b}{c} $
Þ $ \frac{1}{\alpha }+\frac{1}{\beta }={\alpha }’+{\beta }’ $ Hence $ {\alpha }’=\frac{1}{\alpha } $ and $ {\beta }’=\frac{1}{\beta } $ .
BETA
