The point of intersection of lines is `(alpha, beta)` , then the equation whose roots are `alpha, beta`, is

Options

A. `4x^2+x-8=0`
B. `6x^2+x-12=0`
C. `4x^2-x-8=0`
D. `6x^2-x-12=0`

Answer

To find the equation whose roots are the coordinates of the point of intersection of two lines, denoted as \((\alpha, \beta)\), we can use the fact that if \(\alpha\) and \(\beta\) are the roots of a quadratic equation, the equation can be expressed in the standard form:

\[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \]

### Step-by-Step Solution:

1. **Identify the Roots**: The roots of the equation we are looking for are given as \(\alpha\) and \(\beta\).

2. **Sum and Product of Roots**:
- The sum of the roots \(\alpha + \beta\) is represented as \(S\).
- The product of the roots \(\alpha \beta\) is represented as \(P\).

3. **Form the Quadratic Equation**:
- Using the relationship between the roots and coefficients of a quadratic equation, we can write the equation as:
\[ x^2 - Sx + P = 0 \]
where \(S = \alpha + \beta\) and \(P = \alpha \beta\).

4. **Substituting Values**:
- Substitute the values of \(S\) and \(P\) into the equation:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

5. **Final Form**:
- The final form of the equation is:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]

### Conclusion:
The equation whose roots are \(\alpha\) and \(\beta\) is:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]