Complex Numbers And Quadratic Equations question 582
Question: If the sum of the roots of the quadratic equation $ ax^{2}+bx+c=0 $ is equal to the sum of the squares of their reciprocals, then $ \frac{b^{2}}{ac}+\frac{bc}{a^{2}}= $ [BIT Ranchi 1996]
Options:
A) 2
B) -2
C) 1
D) -1
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ \alpha ,\beta $ be the roots of the equation $ ax^{2}+bx+c=0 $ then $ \alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a} $ . Given $ \alpha +\beta =\frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}}\Rightarrow -\frac{b}{a}=\frac{{{\alpha }^{2}}+{{\beta }^{2}}}{{{\alpha }^{2}}{{\beta }^{2}}} $
Þ $ -\frac{b}{a}\frac{c^{2}}{a^{2}}={{(\alpha +\beta )}^{2}}-2\alpha \beta $
Þ $ -\frac{bc^{2}}{a^{3}}=\frac{b^{2}}{a^{2}}-\frac{2c}{a} $
Þ $ \frac{2}{a}=\frac{b^{2}}{a^{2}c}+\frac{bc}{a^{3}}\Rightarrow 2=\frac{b^{2}}{ac}+\frac{bc}{a^{2}} $ .
BETA
