If two equation `a_(1) x^(2) + b_(1) x + c_(1) = 0 and, a_(2) x^(2) + b_(2) x + c_(2) = 0` have a common root, then the value of `(a_(1) b_(2) - a_(2) b_(1)) (b_(1) c_(2) - b_(2) c_(1))`, is

Options

A. `-(a_(1) c_(2) - a_(2) c_(1))^(2)`
B. `(a_(1) a_(2) - c_(1) c_(2))^(2)`
C. `(a_(1) c_(1) - a_(2) c_(2))^(2)`
D. `(a_(1) c_(2) - c_(1) a_(2))^(2)`

Answer

To solve the problem, we need to find the value of the expression \((a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1)\) given that the two quadratic equations \(a_1 x^2 + b_1 x + c_1 = 0\) and \(a_2 x^2 + b_2 x + c_2 = 0\) have a common root.

### Step-by-Step Solution:

1. **Identify the Common Root**:
Let \(r\) be the common root of the two equations. This means:
\[ a_1 r^2 + b_1 r + c_1 = 0 \quad \text{(1)} \]
\[ a_2 r^2 + b_2 r + c_2 = 0 \quad \text{(2)} \]

2. **Multiply the Equations**:
Multiply equation (1) by \(a_2\) and equation (2) by \(a_1\):
\[ a_2(a_1 r^2 + b_1 r + c_1) = 0 \implies a_1 a_2 r^2 + a_2 b_1 r + a_2 c_1 = 0 \quad \text{(3)} \]
\[ a_1(a_2 r^2 + b_2 r + c_2) = 0 \implies a_2 a_1 r^2 + a_1 b_2 r + a_1 c_2 = 0 \quad \text{(4)} \]

3. **Subtract the Two Equations**:
Subtract equation (4) from equation (3):
\[ (a_1 a_2 r^2 + a_2 b_1 r + a_2 c_1) - (a_2 a_1 r^2 + a_1 b_2 r + a_1 c_2) = 0 \]
This simplifies to:
\[ (a_2 b_1 - a_1 b_2)r + (a_2 c_1 - a_1 c_2) = 0 \]

4. **Finding the Value of \(r\)**:
Since \(r\) is a common root, we can express \(r\) as:
\[ r = \frac{a_1 c_2 - a_2 c_1}{a_2 b_1 - a_1 b_2} \quad \text{(5)} \]

5. **Multiply the Equations Again**:
Now, multiply equation (1) by \(b_2\) and equation (2) by \(b_1\):
\[ b_2(a_1 r^2 + b_1 r + c_1) = 0 \implies b_2 a_1 r^2 + b_2 b_1 r + b_2 c_1 = 0 \quad \text{(6)} \]
\[ b_1(a_2 r^2 + b_2 r + c_2) = 0 \implies b_1 a_2 r^2 + b_1 b_2 r + b_1 c_2 = 0 \quad \text{(7)} \]

6. **Subtract the Two New Equations**:
Subtract equation (7) from equation (6):
\[ (b_2 a_1 - b_1 a_2)r^2 + (b_2 c_1 - b_1 c_2) = 0 \]

7. **Finding the Value of \(r^2\)**:
From this, we can express \(r^2\) as:
\[ r^2 = \frac{b_1 c_2 - b_2 c_1}{b_2 a_1 - b_1 a_2} \quad \text{(8)} \]

8. **Equating the Two Expressions**:
From equations (5) and (8), we can equate:
\[ \left(\frac{a_1 c_2 - a_2 c_1}{a_2 b_1 - a_1 b_2}\right)^2 = \frac{b_1 c_2 - b_2 c_1}{b_2 a_1 - b_1 a_2} \]

9. **Final Expression**:
Now, we can multiply both sides by \((a_2 b_1 - a_1 b_2)(b_2 a_1 - b_1 a_2)\) to find:
\[ (a_1 c_2 - a_2 c_1)^2 = (a_2 b_1 - a_1 b_2)(b_1 c_2 - b_2 c_1) \]

Thus, the value of \((a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1)\) is equal to \((a_1 c_2 - a_2 c_1)^2\).