Question

lf $\alpha ,\beta$ are the roots of $x^2+ax-b=0$ and $\gamma ,\delta$ are the roots of $x^2+ax+b=0$, then the value of $(\alpha -\gamma )(\alpha -\delta )$ is:

• A. $2b$
• B. $-2b$
• C. $a+b$
• D. $a-b$

Answer 1

Answer: (A)

$2b$

We know $(\alpha -\gamma )(\alpha -\delta )$ $={\alpha }^2-(\gamma +\delta )\alpha +\gamma .\delta$
Now $\gamma +\delta =-a$ and $\gamma .\delta =b$
Hence, ${\alpha }^2-(\gamma +\delta )\alpha +\gamma .\delta$
$={\alpha }^2+a\alpha +b$ ...(i)
Now $\alpha$ is a root of the equation $x^2+ax-b=0$
Therefore
${\alpha }^2+a\alpha -b=0$
$b={\alpha }^2+a\alpha$
Substituting in (i), we get
${\alpha }^2+a\alpha +b$ $=2b$