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Theory of Equations: Let Roots Equation Least Equals

JEE Maths question with a full step-by-step solution.

Question
Let α\alpha and β\beta be the roots of the equation x2−(p−2)x−(p−1)=0x^2 - (p-2)x - (p-1) = 0, p∈Rp \in \mathbb{R}. If α2+β2\alpha^2 + \beta^2 is least, then pp equals
A00
B11correct
C22
D33
Solution
Step 1: By Vieta's formulas: α+β=p−2\alpha + \beta = p-2 and αβ=−(p−1)\alpha\beta = -(p-1). Step 2:
α2+β2=(α+β)2−2αβ=(p−2)2+2(p−1)=p2−2p+2=(p−1)2+1\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = (p-2)^2 + 2(p-1) = p^2 - 2p + 2 = (p-1)^2 + 1
 Step 3: (p−1)2+1(p-1)^2 + 1 is minimised when p=1p = 1, giving minimum value 11. Correct answer: (2)
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