Complex Numbers And Quadratic Equations question 540
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 $ a/c,b/a,c/b $ are in [AIEEE 2003; DCE 2000]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
As given, if $ \alpha ,\beta $ be the roots of the quadratic equation, then $ \alpha +\beta =\frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}}=\frac{{{(\alpha +\beta )}^{2}}-2\alpha \beta }{{{\alpha }^{2}}{{\beta }^{2}}} $
Þ $ -\frac{b}{a}=\frac{(b^{2}/a^{2})-(2c/a)}{(c^{2}/a^{2})}=\frac{b^{2}-2ac}{c^{2}} $
Þ $ \frac{2a}{c}=\frac{b^{2}}{c^{2}}+\frac{b}{a}=\frac{(ab^{2}+bc^{2})}{ac^{2}} $
Þ $ 2a^{2}c=ab^{2}+bc^{2}\Rightarrow \frac{2a}{b}=\frac{b}{c}+\frac{c}{a} $
Þ $ \frac{c}{a},\frac{a}{b},\frac{b}{c} $ are in A.P.
Þ $ \frac{a}{c},\frac{b}{a},\frac{c}{b} $ are in H.P.
BETA
