$a x^{2} + b x + c = 0$ সমীকরণের মূলদ্বয় $α ও β$ হলে $(a α + b)^{- 2} + (a β + b)^{- 2}$ এর মান নির্ণয় কর।

Created: 2 years ago | Updated: 6 months ago

Updated: 6 months ago

\frac{b^{2} - 2 c a}{a^{2} c^{2}}

Explanation

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Best Ans:

$α$ ও $β$ দুটি মূল, তাই $α + β = \frac{- b}{a}$ এবং $α β = \frac{c}{a}$

এখন, $a α^{2} + b α + c = 0$

$\Rightarrow a α + b = - \frac{c}{α}$

অনুরূপভাবে, $a β + b = \frac{- c}{β}$

আবার, ${(a α + b)}^{- 2} + {(a β + b)}^{- 2} = {(\frac{- c}{α})}^{2} + {(\frac{- c}{β})}^{2}$

$= \frac{α^{2}}{c^{2}} + \frac{β^{2}}{c^{2}} = \frac{α^{2} + β^{2}}{c^{2}} = \frac{{(α + β)}^{2} - 2 α β}{c^{2}}$

$= \frac{\frac{b^{2}}{a^{2}} - 2 \frac{c}{a}}{c^{2}} = \frac{b^{2} - 2 ac}{a^{2} c^{2}}$

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MD.SOLAIMAN HOSAN

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