If the equatoin `ax^(2)-6xy+y^(2)+2bx+2cy+d=0` represents a pair of lines whose slopes are m and `m^(2)`, then value (s) of a is /are

Options

A. `a=-8`
B. `a=8`
C. `a=27`
D. `a=-27`

Explanation

To solve the problem, we need to find the values of \( a \) for the equation \( ax^2 - 6xy + y^2 + 2bx + 2cy + d = 0 \) which represents a pair of lines with slopes \( m \) and \( m^2 \).

### Step-by-Step Solution:

1. **Understanding the Slopes**:
- We are given that the slopes of the lines are \( m \) and \( m^2 \).
- The equations of the lines can be written as:
\[ y - mx = 0 \quad \text{and} \quad y - m^2x = 0 \]

2. **Forming the Combined Equation**:
- The combined equation of the lines can be expressed as:
\[ (y - mx)(y - m^2x) = 0 \]
- Expanding this gives:
\[ y^2 - (m + m^2)xy + m^3x^2 = 0 \]

3. **Comparing Coefficients**:
- The general form of the equation given is:
\[ ax^2 - 6xy + y^2 + 2bx + 2cy + d = 0 \]
- By comparing coefficients, we have:
- Coefficient of \( x^2 \): \( a = m^3 \)
- Coefficient of \( xy \): \( -6 = -(m + m^2) \) which simplifies to \( m + m^2 = 6 \)
- Coefficients of \( x \) and constant term yield \( b = 0 \), \( c = 0 \), and \( d = 0 \).

4. **Finding \( m \)**:
- The equation \( m + m^2 = 6 \) can be rearranged to:
\[ m^2 + m - 6 = 0 \]
- Factoring this quadratic gives:
\[ (m - 2)(m + 3) = 0 \]
- Thus, \( m = 2 \) or \( m = -3 \).

5. **Calculating \( a \)**:
- For \( m = 2 \):
\[ a = m^3 = 2^3 = 8 \]
- For \( m = -3 \):
\[ a = m^3 = (-3)^3 = -27 \]

6. **Final Values of \( a \)**:
- Therefore, the values of \( a \) are:
\[ a = 8 \quad \text{or} \quad a = -27 \]

### Conclusion:
The values of \( a \) that satisfy the given conditions are \( a = 8 \) and \( a = -27 \).

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