Sequence And Series Question 298
Question: If in the equation $ ax^{2}+bx+c=0, $ the sum of roots is equal to sum of square of their reciprocals, then $ \frac{c}{a},\frac{a}{b},\frac{b}{c} $ are in
[RPET 2000]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ \alpha +\beta =\frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}}= $ $ \frac{{{\alpha }^{2}}+{{\beta }^{2}}}{{{(\alpha ,\beta )}^{2}}} $ $ =\frac{{{(\alpha +\beta )}^{2}}-2\alpha \beta }{{{(\alpha ,\beta )}^{2}}} $ ?..(i) $ \alpha +\beta =-b/a $ and $ \alpha \beta =c/a $ Putting these value in (i)
Þ $ ( \frac{-b}{a} ),( \frac{c^{2}}{a^{2}} )=\frac{b^{2}}{a^{2}}-\frac{2c}{a} $ or $ -bc^{2}=ab^{2}-2ca^{2} $ or $ 2c,a^{2}=ab^{2}+bc^{2} $ Dividing by abc we get, $ \frac{2a}{b}=\frac{b}{c}+\frac{c}{a} $
Þ $ \frac{c}{a},\frac{a}{b},\frac{b}{c} $ are in A.P.
BETA
