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Consistency of Linear System of Equations

If α,β are ...

Question

If $α, β$ are the roots of ${a x}^{2} + b x + c = 0, (a \neq 0)$ and $α + δ, β + δ$ are the roots of the equation ${A x}^{2} + B x + C = 0, (A \neq 0)$ for the same constant $δ$ , then

A

\frac{b^{2} - 4 a c}{A^{2}} = \frac{B^{2} - 4 A C}{a^{2}}

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B

\frac{b^{2} - 4 a c}{a^{2}} = \frac{B^{2} - 4 A C}{A^{2}}

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C

\frac{b^{2} - 4 a c}{b^{2}} = \frac{B^{2} - 4 A C}{A^{2}}

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D

\frac{b^{2} - 4 a c}{B^{2}} = \frac{B^{2} - 4 A C}{a^{2}}

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Solution

The correct option is B $\frac{b^{2} - 4 a c}{a^{2}} = \frac{B^{2} - 4 A C}{A^{2}}$
$α, β$ be the roots of $a x^{2} + b x + c = 0$ and $α + δ, β + δ$ are roots of $A x^{2} + B x + C = 0$
we know that difference between the roots
$| α - β | = \frac{\sqrt{b^{2} - 4 a c}}{a}$
similarly $| α + δ - β - δ | = \frac{\sqrt{B^{2} - 4 A C}}{A}$
$∴ \frac{b^{2} - 4 a c}{a^{2}} = \frac{B^{2} - 4 A C}{A^{2}}$

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