SATHEE: Theory of Equations 1 Question 45

Theory of Equations 1 Question 45

46. 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 $A x^{2} + B x + C = 0, (A \neq 0)$ for some constant $δ$ , then prove that

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

$(2000, 4 M)$

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Solution:

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

$and α + δ + β + δ = - \frac{B}{A}, (α + δ) (β + δ) = \frac{C}{A}$

$Now, α - β = (α + δ) - (β + δ)$

$\begin{array}{ll} \Rightarrow & (α - β)^{2} = [(α + δ) - (β + δ)]^{2} \\ \Rightarrow & (α + β)^{2} - 4 α β = (\overset{―}{α + δ} + \overset{―}{β + δ)^{2} - 4 (α + δ) (β + δ)} \\ \Rightarrow & - {\frac{b}{a}}^{2} - \frac{4 c}{a} = - \frac{B^{2}}{A} - \frac{4 C}{A} \\ \Rightarrow & \frac{b^{2}}{a^{2}} - \frac{4 c}{a} = \frac{B^{2}}{A^{2}} - \frac{4 C}{A} \\ \Rightarrow & \frac{b^{2} - 4 a c}{a^{2}} = \frac{B^{2} - 4 A C}{A^{2}} \end{array}$

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Theory of Equations 1 Question 45

46. If α,β are the roots of ax2+bx+c=0,(a≠0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A≠0) for some constant δ, then prove that

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46. 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 $A x^{2} + B x + C = 0, (A \neq 0)$ for some constant $δ$ , then prove that