QJEEM21Ma7dbc: JEE Main 2021 Mathematics - Class 11 JEEM21Ma7dbc Solution: Let a, b, c be in arithmetic progression. Let the centroid of…

Question

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Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be

\left( {{{10} \over 3},{7 \over 3}} \right)

. If

\alpha

,

\beta

are the roots of the equation

a{x^2} + bx + 1 = 0

, then the value of

{\alpha ^2} + {\beta ^2} - \alpha \beta

is :

Accepted Answer

2b = a + c

{{2a + 2} \over 3} = {{10} \over 3}

and

{{2b + c} \over 3} = {7 \over 3}

a = 4,

{\begin{matrix} 2 b + c = 7 \\ 2 b - c = 4 \end{matrix}}

, solving

b = {{11} \over 4}

c = {3 \over 2}

\therefore

Quadratic Equation is

4{x^2} + {{11} \over 4}x + 1 = 0

\therefore

The value of

{\alpha ^2} + {\beta ^2} - \alpha \beta

=

{\alpha ^2} + {\beta ^2} + 2\alpha \beta - 3\alpha \beta

=

{(\alpha + \beta )^2} - 3\alpha \beta

= {{121} \over {256}} - {3 \over 4} = - {{71} \over {256}}

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