Question

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

• A. $pqr$
• B. $2pqr$
• C. $p^2{q}^2{r}^2$
• D. None of these

Answer 1

The correct option is A $pqr$

Let the roots of $x^2+px+qr=0$ be $\alpha$ and $\beta$

$\therefore \alpha \beta =qr$ ...(1)

Let the roots of $x^2+qx+rp=0$ be $\beta$ and $\gamma$

$\therefore \beta \gamma =rp$ ...(2)

Let the roots of $x^2+rx+pq=0$ be $\alpha$ and $\gamma$

$\therefore \alpha \gamma =pq$ ...(3)

Multiplying equation $\left(1\right),\left(2\right)$ and $\left(3\right)$, we get

$\therefore {\alpha }^2{\beta }^2{\gamma }^2=p^2{q}^2{r}^2$

$\therefore \alpha \beta \gamma =pqr$