# The value of $'a'$ for which sum of squares of roots of $x^{2}-\left ( a-2 \right )x-\left ( a+1 \right )= 0$  is minimum is Option 1) -1 Option 2) 0 Option 3) 1 Option 4) 2

P Plabita

Let $\alpha ,\, \beta$ are roots of given equation.

$\alpha ^{2}+\beta ^{2}=\left ( \alpha +\beta \right )^{2}-2\alpha \beta =\left ( a-2 \right )^{2}+2\left ( a+1 \right )$

$\Rightarrow \: \alpha ^{2}+\beta ^{2}=a^{2}-2a+6= \left ( a-1 \right )^{2}+5$

$\Rightarrow \: \alpha ^{2}+\beta ^{2}$  will be minimum when $\left ( a-1 \right )^{2}=0$

$\Rightarrow \: a=1$

$\therefore$ Option (C)

Product of Roots in Quadratic Equation -

$\alpha \beta = \frac{c}{a}$

- wherein

$\alpha \: and\ \beta$ are roots of quadratic equation:

$ax^{2}+bx+c=0$

$a,b,c\in C$

Option 1)

-1

This is incorrect

Option 2)

0

This is incorrect

Option 3)

1

This is correct

Option 4)

2

This is incorrect

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