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If the equation x2 + kx + 1 = 0 has the roots α and β, then what will be the equation with the roots \(-\left(\frac{\alpha \beta + 1}{\beta}\right)\) and \(-\left(\frac{\alpha \beta + 1}{\alpha}\right)\) ?

If the equation x2 + kx + 1 = 0 has the roots α and β, then what will be the equation with the roots \(-\left(\frac{\alpha \beta + 1}{\beta}\right)\) and \(-\left(\frac{\alpha \beta + 1}{\alpha}\right)\) ? Correct Answer x<sup>2</sup> - 2kx + 4 = 0

Given:

α and β is the roots of x2 + kx + 1 = 0

Concept:

 α + β = -b/a, α × β = c/a Where α and β is the root of the equation x2 + bx + c = 0

Calculation:

⇒ α + β = -k/1, α × β = 1

⇒ Sum of roots = -(αβ + 1)/β  - (αβ + 1)/α = -2/β - 2/α = -2 × (α + β)/(α × β) = -2 × (-k) = 2k

⇒ product of roots =   {-(αβ + 1)/β} × {-(αβ + 1)/α} = 4/(α × β)  = 4

⇒ -b/a = 2k/1, and c/a = 4/1

⇒ The equation x2 - 2kx + 4 = 0 is shows such relation

∴ The required result will be "x2 - 2kx + 4 = 0".

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