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Algebra of Limits

if α and ...

Question

if $α$ and $β$ are the roots of the equation $a x^{2} + b x + c = 0$ then the sum of the roots of the equation $a^{2} x^{2} + (b^{2} - 2 a c) x + b^{2} - 4 a c = 0$ is

A

- (α^{2} - β^{2})

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B

{(α + β)}^{2} - 2 α β

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C

α^{2} β + β α^{2} - 4 α β

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D

- (α^{2} + β^{2})

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Solution

The correct option is D $- (α^{2} + β^{2})$
As $α, β$ are the roots of equation $a x^{2} + b x + c = 0$

then
$α + β = \frac{- b}{a}$
$α β = \frac{c}{a}$

The sum of the roots of the equation is given by,
sum of roots=
$- \frac{b^{2} - 2 a c}{a^{2}} = - (\frac{b}{a})^{2} + 2 \frac{c}{a} ⟹ = - (α + β)^{2} + 2 α β ⟹ = - (α^{2}) - β^{2} - 2 α β + 2 α β ⟹ - (α^{2} + β^{2})$

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