If each pair of the following equations `x^2+px+qr=0, x^2+qx+pr=0 and x^2+rx+pq=0 ` has common root, then the product of the three common roots is (A) `2pqr` (B) `pqr` (C) `-pqr` (D) none of these

Options

A. 2pqr
B. pqr
C. `-pqr`
D. none of these

Answer

To solve the problem, we need to find the product of the three common roots of the given quadratic equations. Let's denote the equations as follows:

1. \( x^2 + px + qr = 0 \) (Equation 1)
2. \( x^2 + qx + pr = 0 \) (Equation 2)
3. \( x^2 + rx + pq = 0 \) (Equation 3)

We know that each pair of these equations has a common root. Let's denote the common roots as follows:

- Let \( \alpha \) be the common root of Equation 1 and Equation 2.
- Let \( \beta \) be the common root of Equation 2 and Equation 3.
- Let \( \gamma \) be the common root of Equation 1 and Equation 3.

### Step 1: Find the product of the roots for each equation.

Using Vieta's formulas, we can express the product of the roots for each equation:

- For Equation 1, the product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{qr}{1} = qr \]

- For Equation 2, the product of the roots \( \alpha \gamma \) is given by:
\[ \alpha \gamma = \frac{c}{a} = \frac{pr}{1} = pr \]

- For Equation 3, the product of the roots \( \beta \gamma \) is given by:
\[ \beta \gamma = \frac{c}{a} = \frac{pq}{1} = pq \]

### Step 2: Relate the products of the roots.

Now, we have the following relationships:
1. \( \alpha \beta = qr \)
2. \( \alpha \gamma = pr \)
3. \( \beta \gamma = pq \)

### Step 3: Multiply all three equations.

To find the product of the three common roots \( \alpha \), \( \beta \), and \( \gamma \), we can multiply the three equations together:

\[ (\alpha \beta)(\alpha \gamma)(\beta \gamma) = (qr)(pr)(pq) \]

This simplifies to:

\[ \alpha^2 \beta^2 \gamma^2 = p^2 q^2 r^2 \]

### Step 4: Take the square root.

Taking the square root of both sides gives us:

\[ \alpha \beta \gamma = \pm pqr \]

### Conclusion:

The product of the three common roots \( \alpha \beta \gamma \) is \( \pm pqr \). Therefore, the answer is:

- (B) \( pqr \) or (C) \( -pqr \)