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If the roots of the equations $x^{2}-(a-1) x+(a+b)=0$ and $a x^{2}-2 x+b=0$ are identical, then what are the values of a and $\mathrm{b}$ ?
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Verified Answer
The correct answer is:
$a=2, b=-4$
Let $\alpha$ and $\beta$ be the roots of both the equations $x^{2}-(a-1) x+(a+b)=0$
$\Rightarrow \alpha+\beta=(a-1)$ and $\alpha \beta=(a+b)$
and $a x^{2}-2 x+b=0$
$\Rightarrow \alpha+\beta=\frac{2}{a}$ and $\alpha \beta=\frac{b}{a}$
Equating the sums of roots
$a^{2}-a-2=0 \Rightarrow a=-1,2$
Equating the products of roots and
and if $a=2, b=$ $1=$
From the given option, $=-4$ matches.
$\Rightarrow \alpha+\beta=(a-1)$ and $\alpha \beta=(a+b)$
and $a x^{2}-2 x+b=0$
$\Rightarrow \alpha+\beta=\frac{2}{a}$ and $\alpha \beta=\frac{b}{a}$
Equating the sums of roots
$a^{2}-a-2=0 \Rightarrow a=-1,2$
Equating the products of roots and
and if $a=2, b=$ $1=$
From the given option, $=-4$ matches.
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