Complex Numbers And Quadratic Equations question 601
Question: The value of $ k $ for which one of the roots of $ x^{2}-x+3k=0 $ is double of one of the roots of $ x^{2}-x+k=0 $ is [UPSEAT 2001]
Options:
A) 1
B) - 2
C) 2
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ \alpha $ be a root of $ x^{2}-x+k=0, $ then $ 2\alpha $ is a root of $ x^{2}-x+3k=0 $ .
$ \therefore {{\alpha }^{2}}-\alpha +k=0 $ and $ 4{{\alpha }^{2}}-2\alpha +3k=0 $
Þ $ \frac{{{\alpha }^{2}}}{-3k+2k}=\frac{\alpha }{4k-3k}=\frac{1}{-2+4} $
Þ $ {{\alpha }^{2}}=-k/2 $ and $ \alpha =k/2 $ Now, $ {{\alpha }^{2}}={{(\alpha )}^{2}}\Rightarrow -k/2={{(k/2)}^{2}} $
$ \Rightarrow k^{2}+2k=0\Rightarrow k=0 $ or - 2.
BETA
