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 $ab=\alpha \beta +5\dots \left(i\right)$
Taking another equation $\left(x-\alpha \right)\left(x-\beta \right)+5=0$
$\Rightarrow x^2-\left(\alpha +\beta \right)x+\left(\alpha \beta +5\right)=0$
$\Rightarrow x^2-\left(a+b\right)x+ab=0$ (using(i))
$\therefore$ Its roots are $a,b$.