Question

If $\alpha ,\beta$ be the roots of $x^2+px+1=0$ and $\gamma ,\delta$ be the roots of $x^2+qx+1=0$, then $(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )=$

• A. $p^2+q^2$
• B. $p^2-q^2$
• C. $q^2-p^2$
• D. None of these

Answer 1

The correct option is C $q^2-p^2$

Here, $\alpha +\beta =p;\alpha \beta =1,\gamma +\delta =q;\gamma \delta =1$
$\Rightarrow \alpha \beta =\gamma \delta$
Now,$(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )$
$=\{\alpha \beta -\gamma (\alpha +\beta )+{\gamma }^2\}\{\alpha \beta +\delta (\alpha +\beta )+{\delta }^2\}$
$=\{1+\gamma p+{\gamma }^2\}\{1-p\delta +{\delta }^2\}$ $=\{({\gamma }^2+1)+\gamma p\}\{({\delta }^2+1)-p\delta \}$
$=(-q\gamma +\gamma p)(-q\delta -p\delta )$ $=\gamma \delta (q^2-p^2)=1(q^2-p^2)$