Complex Numbers And Quadratic Equations question 176
Question: If one root of the equation $ ax^{2}+bx+c=0 $ be $ n $ times the other root, then
Options:
A) $ na^{2}=bc{{(n+1)}^{2}} $
B) $ nb^{2}=ac{{(n+1)}^{2}} $
C) $ nc^{2}=ab{{(n+1)}^{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Let the roots are $ \alpha $ and $ n\alpha $ Sum of roots $ \alpha +n\alpha =-\frac{b}{a} $
Þ $ \alpha =-\frac{b}{a(n+1)} $ …..(i) and product, $ \alpha .n\alpha =\frac{c}{a} $
Þ $ {{\alpha }^{2}}=\frac{c}{na} $ ….(ii) From (i) and (ii), we get
Þ $ {{[ -\frac{b}{a(n+1)} ]}^{2}}=\frac{c}{na} $
Þ $ \frac{b^{2}}{a^{2}{{(n+1)}^{2}}}=\frac{c}{na} $
$ \Rightarrow $ $ nb^{2}=ac{{(n+1)}^{2}} $ . Note: Students should remember this question as a fact.
BETA
