If two of the straight lines respresented by the `3x^3+3x^2y-3xy^2+ay^3=0` are at right angles .then the slope of one them is

Options

A. `-1`
B. 1
C. 3
D. `-3`

Answer

To solve the problem, we need to analyze the given cubic equation and determine the slopes of the straight lines represented by it, particularly focusing on the condition that two of the lines are perpendicular to each other.

### Step-by-Step Solution:

1. **Given Equation**: Start with the given equation of the cubic polynomial:
\[ 3x^3 + 3x^2y - 3xy^2 + ay^3 = 0 \]

2. **Condition for Right Angles**: For two lines to be at right angles, the product of their slopes must equal -1. This can be derived from the general form of a cubic equation representing three lines.

3. **Rearranging the Equation**: We can express the equation in the form:
\[ 3(x^3 + x^2y - xy^2 + \frac{a}{3}y^3) = 0 \]
This suggests that we can factor it into linear factors.

4. **Factoring the Polynomial**: We can assume the lines can be represented as:
\[ (x^2 + py - q)(x + ry + s) = 0 \]
where \(p\), \(q\), \(r\), and \(s\) are constants that we will determine.

5. **Using the Right Angle Condition**: Since two of the lines are perpendicular, we can set up the condition:
\[ p \cdot r = -1 \]
This means that if one slope is \(m_1\) and the other is \(m_2\), then \(m_1 \cdot m_2 = -1\).

6. **Comparing Coefficients**: By expanding the assumed factorization and comparing coefficients with the original polynomial, we can derive relationships between \(a\), \(p\), and \(r\).

7. **Setting Up the System of Equations**: From the coefficients of \(x^3\), \(x^2y\), \(xy^2\), and \(y^3\), we can derive:
- Coefficient of \(x^2y\): \(3 = p + r\)
- Coefficient of \(xy^2\): \(-3 = -pr\)
- Coefficient of \(y^3\): \(a = 0\)

8. **Solving for \(p\) and \(r\)**: Substitute \(a = -3\) into the equations and solve for \(p\) and \(r\):
- From \(p + r = 3\) and \(pr = 3\), we can solve these equations to find the values of \(p\) and \(r\).

9. **Finding the Slopes**: The slopes of the lines can be derived from the values of \(p\) and \(r\). If we assume \(p = 1\) and \(r = -3\), then the slopes of the lines are:
- Slope 1: \(m_1 = 1\)
- Slope 2: \(m_2 = -1\)

10. **Conclusion**: The slopes of the lines represented by the equation are \(1\) and \(-1\). Therefore, the slope of one of the lines is:
\[ \text{Slope} = 1 \text{ or } -1 \]