Question

If every pair from among the equations $x^2+px+qr=0,x^2+qx+rp=0$ and $x^2+rx+pq$ = 0 has a common root, then the sum of the three common roots is

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

Answer 1

Answer: (B)

The given equations are
$x^2+px+qr=0$ $...(1)$
$x^2+qx+rp=0$ $...(2)$
$x^2+rx+pq=0$ $...(3)$
Let $\alpha ,\beta$ be the roots of (1), $\beta ,\gamma$ be the roots of $(2)$ and $\gamma ,\alpha$ be the roots of $(3)$.
Since $\beta$ is a common root of $(1)$ and $(2)$.
$\therefore {\beta }^2+p\beta +qr=0$ $...(4)$
${\beta }^2+q\beta +rp=0$ $....(5)$
(4) - (5) gives
$(p-q)\beta -(q-p)r=0$
$\therefore \beta =r$
Now, $\alpha \beta =qr$
$\Rightarrow ar=qr$
$\Rightarrow \alpha =q$
Again $\beta \gamma =rp$
$\therefore r\gamma =rp$
$\therefore \gamma =p$
$\therefore \alpha +\beta +\gamma =q+r+p$
$=p+q+r$