Complex Numbers And Quadratic Equations question 546
Question: If the roots of the equation $ Ax^{2}+Bx+C=0 $ are $ \alpha ,\beta $ and the roots of the equation $ x^{2}+px+q=0 $ are $ {{\alpha }^{2}},\ {{\beta }^{2}} $ , then value of p will be [RPET 1986]
Options:
A) $ \frac{B^{2}-2AC}{A^{2}} $
B) $ \frac{2AC-B^{2}}{A^{2}} $
C) $ \frac{B^{2}-4AC}{A^{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ \alpha ,\beta $ are the roots of $ Ax^{2}+Bx+C=0 $ . So, $ \alpha +\beta =-\frac{B}{A} $ and $ \alpha \beta =\frac{C}{A} $ Again $ {{\alpha }^{2}},{{\beta }^{2}} $ are the roots of $ x^{2}+px+q=0 $ then $ {{\alpha }^{2}}+{{\beta }^{2}}=-p $ and $ {{(\alpha \beta )}^{2}}=q $ Now $ {{\alpha }^{2}}+{{\beta }^{2}}={{(\alpha +\beta )}^{2}}-2\alpha \beta $
Þ $ {{\alpha }^{2}}+{{\beta }^{2}}={{( -\frac{B}{A} )}^{2}}-2\frac{C}{A} $
Þ $ -p=\frac{B^{2}-2AC}{A^{2}}\Rightarrow p=\frac{2AC-B^{2}}{A^{2}} $
BETA
