Question

If every pair from among the equations $x^2+px+qr+qx+rp=0and{x}^2+rx+pq=0$have a common root, then $[$sum of roots $/$product of roots $]$ is

• A. $\sum p/pqr$
• B. $i/pqr$
• C. ${(p+q+r)}^2$
• D. None of these

Answer 1

The correct option is A

$\sum p/pqr$

Explanation for the correct option:

Step 1. Find the value of $[$sum of roots $/$product of roots $]$

Given that the equations $x^2+px+qr+qx+rp=0and{x}^2+rx+pq=0$have a common root

Let $\alpha ,\beta$ be the roots of $x^2+px+qr=0—\left(1\right)$and

$\beta ,\gamma$ be the roots of $x^2+qx+rp=0—\left(2\right)$and

$\gamma ,\alpha$ be the roots of $x^2+rx+pq=0—\left(3\right)$

Step 2. Find the sum of the roots:

Since $\beta$ is a common root of $\left(1\right),\left(2\right)$

${\beta }^2+p\beta +qr=0$and ${\beta }^2+q\beta +rp=0$

$\Rightarrow$$\beta –r=0$

$\Rightarrow$ $\beta =r$

Again,

$ɑ\beta =qr$

$\Rightarrow$$ɑr=qr$

$\Rightarrow$ $ɑ=q$

Similarly from $\left(2\right)𝜸=p$, from $\left(3\right)$ and $\left(1\right),\beta =r$

$\Rightarrow$ $ɑ+\beta +𝜸=q+r+p=p+q+r$

Step 3. Find the product of the roots:

$\left(ɑ\beta \right)(\beta 𝜸)(𝜸ɑ)=\left(qr\right)\left(rp\right)\left(pq\right)$

$\Rightarrow$ ${(ɑ\beta 𝜸)}^2={\left(pqr\right)}^2$

$\Rightarrow$ $ɑ\beta 𝜸=pqr$

$\therefore [sumoftheroots/productoftheroots]=\left[\right(ɑ\beta )+(\beta 𝜸)+(𝜸ɑ\left)\right]/ɑ\beta 𝜸$

$\mathbf{[}\mathit{p}\mathbf{+}\mathit{q}\mathbf{+}\mathit{r}\mathbf{]}\mathbf{/}\mathit{p}\mathit{q}\mathit{r}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\sum }\mathit{p}\mathbf{}\mathbf{/}\mathbf{}\mathit{p}\mathit{q}\mathit{r}$

Hence, Option ‘A’ is Correct.