Question

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

• A. $(-q^2-p^2)$
• B. $(p^2-q^2)$
• C. $(q^2-p^2)$
• D. $(q^2+p^2)$

Answer 1

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

$x^2+px+1=0$

Sum and product of roots,
$\alpha +\beta =-p,\alpha \beta =1$

$x^2+qx+1=0$

Sum and product of roots,
$\gamma +\delta =-q,\gamma \delta =1$

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

We know that,
$x^2+qx+1=0$ has roots as $\gamma ,\delta$

So,
$(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )=\left[\right({\gamma }^2+1)+\gamma p]\left[\right({\delta }^2+1)-p\delta )]=(-q\gamma +\gamma p\left)\right(-q\delta -p\delta )=\gamma \delta (q^2-p^2)=(q^2-p^2)$