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$