Question

If $\alpha, \beta$ are the roots of the equation $(x-a)(x-b)=5$, then the roots of the equation $\left(x-\alpha\right)\left(x-\beta\right)+5=0$ are

• A. $a, 5$
• B. $b, 5$
• C. $a, \alpha$
• D. $a, b$

Answer 1

Answer: (D)

Since $\alpha, \beta$ are roots of the equation
$\left(x-a\right)\left(x-b\right)=5$
or $x^{2}-\left(a+b\right)x+\left(ab-5\right)=0$
$\therefore \, \alpha+\beta=a+b$ or $a+b=\alpha+\beta$
and $ \alpha\beta=ab-5$ or $ab=\alpha\beta+5\quad\ldots\left(i\right)$
Taking another equation $\left(x-\alpha\right)\left(x-\beta\right)+5=0$
$\Rightarrow \, x^{2}-\left(\alpha+\beta\right)x+\left(\alpha\beta+5\right)=0$
$\Rightarrow \, x^{2}-\left(a+b\right)x+ab=0\quad$ (using(i))
$\therefore $ Its roots are $a, b$.