First slide

Theory of equations

Question

$If the roots of the quadratic equation a x^{2} + b x - b = 0; where a, b \in R, such that a \cdot b > 0, are α and β then the$ $value of \log_{|β - 1|} |α - 1| is$

Moderate

A

1
B

-1
C

0
D

None of these

Solution

$Roots of a x^{2} + b x - b = 0 are α and β$
$\begin{array}{l} ∴ α + β = \frac{- b}{a} and α \cdot β = \frac{- b}{a} \\ \Rightarrow α \cdot β - α - β = 0 \\ \Rightarrow α \cdot β - α - β + 1 = 0 + 1 \\ \Rightarrow (α - 1) (β - 1) = 1 \\ \Rightarrow | α - 1 | \cdot | β - 1 | = 1 \\ \Rightarrow | α - 1 | = \frac{1}{| β - 1 |} \\ \Rightarrow \log | α - 1 | = \log \frac{1}{| β - 1 |} \\ \Rightarrow \log | α - 1 | = l og 1 - \log |β - 1| \\ \Rightarrow \log | α - 1 | = - \log |β - 1| \\ \Rightarrow \log_{| (β - 1) |} ∣ α - 1 ∣= - 1 \end{array}$

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