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Quantitative Aptitude

Maxima/Minima of a Quadratic Equation

If p and q ar...

Question

If p and q are the roots of the quadratic equation $x^{2} - (α - 2) x - α - 1 = 0$ . What is the minimum possible value of $p^{2} + q^{2}$ ?

A

0

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B

3

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C

4

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D

5

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Solution

The correct option is D 5
$x^{2} - (α - 2) - α - 1 = 0$
Then sum of roots= $α - 2$
product of roots = $- α - 1$
$(p + q)^{2} = p^{2} + q^{2} + 2 p q$
Thus $(α - 2)^{2} = p^{2} + q^{2} + 2 (- α - 1)$
$p^{2} + q^{2} = α^{2} - 4 α + 4 + 2 α + 2 = α^{^{2}} - 2 α + 6 = α^{2} - 2 α + 1 + 5 = (α - 1)^{2} + 5$
Thus minimum velue of $p^{2} + q^{2}$ is 5

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