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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
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B
GP
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C
HP
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D
None of these
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Solution

The correct option is B HP
Given, 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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