For what value of k will the roots of the equation kx^2 – 5x + 6 = 0 be in the ratio 2 : 3 ? -Maths 9th

For what value of k will the roots of the equation kx^2 – 5x + 6 = 0 be in the ratio 2 : 3 ? -Maths 9th
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Answer :

(b) 1 Let the roots of the equation kx2 – 5x + 6 = 0 be α and β. Then, α + β = \(\frac{5}{k}\)                ...(i) αβ = \(\frac{6}{k}\)                                 ...(ii) Given \(\frac{α}{​​β}\) = \(\frac{2}{3}\) ⇒ α = \(\frac{2}{3}\)β ∴ From (i) and (ii),   \(\frac{2}{3}\)β + β = \(\frac{5}{k}\) and \(\frac{2}{3}\)β2 = \(\frac{6}{k}\) ⇒ \(\frac{5}{3}\)β = \(\frac{5}{k}\) and  β2 = \(\frac{9}{k}\) ⇒ β = \(\frac{3}{k}\) and  β2 = \(\frac{9}{k}\) ⇒ \(\frac{9}{k^2}\) = \(\frac{9}{k}\) ⇒ 9k2 - 9k = 0 k(k - 1) = 0 ⇒ k = 0 or 1 But k = 0 does not satisfy the condition, so k = 1.
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