If α, ß are the roots of the equation 3x2 – 4x – 6 = 0, find the equation whose roots are α2 + 1 and ß2 + 1.
Explanation:
Since α, ß are roots of the equation 3x2 – 4x – 6 = 0
∴ α + ß = 43 and αß = - 63 = - 2
Now QE whose roots are α2 + 1 and ß2 + 1 can be formed as
x2 - (sum of the roots)x + (product of the roots) ∴ x2 – (α2 + 1 + ß2 + 1) + (α2 + 1)(ß2 + 1)
Now, ⇒ α2 + 1 + ß2 + 1 = (α + ß)2 – 2αß + 2 = 169 + 4 + 2 = 169 + 6 = 709
and (α2 + 1)(ß2 + 1) = α2ß2 + (α2 + ß2) + 1 = (αß)2 + (α + ß)2 - 2αß + 1 = 4 + 169 + 4 + 1 = 169 + 9 = 979
∴ The required equation is x2 - 709x + 979 = 0
⇒ 9x2 - 70x + 97 = 0
Hence, option (b).
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