Question

# The sum of the roots of quadratic equation $$ax^2 + bx + c = 0 \,(a, b, \neq 0)$$ is equal to the sum of squares of their reciprocals, then $$\dfrac {a}{c}, \dfrac{b}{a}$$ and $$\dfrac {c}{b}$$ are in

A
AP
B
GP
C
HP
D
None of these

Solution

## The correct option is B HPGiven, equation is $$ax^2+bx+c=0$$If $$\alpha$$ and $$\beta$$ be the roots of this equation.Then, according to question$$\alpha +\beta =\dfrac{1}{\alpha ^2}+\dfrac {1}{\beta ^2}=\dfrac{\alpha ^2+\beta ^2}{\alpha ^2\beta ^2}$$$$=\dfrac{(\alpha +\beta )^2-2\alpha \beta }{\alpha ^2\beta^2 }$$$$\Rightarrow \dfrac{-b}{a}=\dfrac {b^2-2ac}{c^2}$$$$\Rightarrow \dfrac{-b}{a}=\dfrac{b^2}{c^2}+\dfrac{b}{a}=\dfrac{ab^2+bc^2}{ac^2}$$$$\Rightarrow 2a^2c=ab^2+bc^2$$$$\Rightarrow \dfrac {2a}{b}+\dfrac{b}{c}+\dfrac {c}{a}$$Here $$\dfrac {c}{a},\dfrac{a}{b}$$ and $$\dfrac {b}{c}$$ are in AP.Therefore, $$\dfrac {c}{a},\dfrac{b}{a}$$ and $$\dfrac {c}{b}$$ are in HP.Maths

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