Question

If $\alpha ,\beta$ are the roots of $x^2+px+q=0$, and $\gamma ,\delta$ are the roots of $x^2+rx+s=0$, evaluate $(\alpha -\gamma )(\alpha -\delta )(\beta -\gamma )(\beta -\delta )$ in terms of $p,q,r$ and $s$.

• A. ${(r-s)}^2+(p-q)(rq-sp)$
• B. ${(p-s)}^2+(q-r)(rq-sp)$
• C. ${(q-s)}^2+(p-r)(rq-sp)$
• D. ${(s-r)}^2+(p-q)(rq-sp)$

Answer 1

The correct option is C ${(q-s)}^2+(p-r)(rq-sp)$

$\therefore \alpha ,\beta$ are the roots of $x^2+px+q=0$
$\therefore \alpha +\beta =-p,\alpha \beta =q$ .............(1)
and $\gamma ,\delta$ are the roots of $x^2+rx+s=0$
$\therefore \gamma +\delta =-r,\gamma \delta =s$............(2)
Now,
$(\alpha -\gamma )(\alpha -\delta )(\beta -\gamma )(\beta -\delta )$
$=[{\alpha }^2-\alpha (\gamma +\delta )+\gamma \delta ][{\beta }^2-\beta (\gamma +\delta )+\gamma \delta ]$
$=({\alpha }^2+r\alpha +s)({\beta }^2+r\beta +s)$ {from(2)}
$={\alpha }^2{\beta }^2+r\alpha \beta (\alpha +\beta )+r^2\alpha \beta +s({\alpha }^2+{\beta }^2)+sr(\alpha +\beta )+s^2$
$={\alpha }^2{\beta }^2+r\alpha \beta (\alpha +\beta )+r^2\alpha \beta +s({(\alpha +\beta )}^2-2\alpha \beta +sr(\alpha +\beta )+s^2)$
$=q^2-pqr+r^{2}q+s(p^2-2q)+sr(-p)+s^2$
$={(q-s)}^2-rpq+r^{2}q+s{p}^2-prs$
$={(q-s)}^2-rq(p-r)+sp(p-r)$
$={(q-s)}^2+(p-r)(sp-rq)$ ............(3)
For a common root (let $\alpha =\gamma or\beta =\delta$ )
$(\alpha -\gamma )(\alpha -\delta )(\beta -\gamma )(\beta -\delta )=0$ .......(4)
from (3) &(4), we get ${(q-s)}^2+(p-r)(sp-rq)=0$
$\Rightarrow {(q-s)}^2+(p-r)(rq-sp)$, which is required condition.