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 find the slope of one of the lines represented by the equation \(3x^3 + 3x^2y - 3xy^2 + ay^3 = 0\) under the condition that two of these lines are perpendicular to each other.

### Step-by-step Solution:

1. **Identify the Equation**:
The given equation is:
\[ 3x^3 + 3x^2y - 3xy^2 + ay^3 = 0 \]

2. **Assume the Slopes**:
Let the slopes of the lines represented by the equation be \(m_1\), \(m_2\), and \(m_3\). The product of the slopes can be derived from the coefficients of the equation.

3. **Use Vieta's Formulas**:
According to Vieta's formulas, for a cubic equation \(ax^3 + bx^2 + cx + d = 0\), the product of the roots (slopes in this case) is given by:
\[ m_1 m_2 m_3 = -\frac{d}{a} \]
Here, \(d = a\) and \(a = 3\), thus:
\[ m_1 m_2 m_3 = -\frac{a}{3} \]

4. **Condition for Perpendicular Lines**:
If two lines are perpendicular, the product of their slopes is \(-1\). Without loss of generality, let’s assume \(m_1\) and \(m_2\) are the slopes of the perpendicular lines:
\[ m_1 m_2 = -1 \]
Therefore, we can express \(m_3\) in terms of \(m_1\) and \(m_2\):
\[ m_3 = -\frac{1}{m_1} \quad \text{(since \(m_1 m_2 = -1\))} \]

5. **Substituting into Vieta's Formula**:
Substitute \(m_3\) into the product of the slopes:
\[ m_1 m_2 m_3 = m_1 m_2 \left(-\frac{1}{m_1}\right) = -m_2 \]
Thus, we have:
\[ -m_2 = -\frac{a}{3} \]
This implies:
\[ m_2 = \frac{a}{3} \]

6. **Finding the Value of \(a\)**:
From the earlier steps, we know:
\[ m_1 m_2 = -1 \quad \Rightarrow \quad m_1 \cdot \frac{a}{3} = -1 \quad \Rightarrow \quad m_1 = -\frac{3}{a} \]
Now substituting \(m_1\) into the product of slopes:
\[ -\frac{3}{a} \cdot \frac{a}{3} \cdot m_3 = -\frac{a}{3} \]
This leads to:
\[ m_3 = \frac{3}{a} \]

7. **Final Calculation**:
We already have \(m_1 = -\frac{3}{a}\) and \(m_2 = \frac{a}{3}\). The slopes of the lines are:
\[ m_1 = -1, \quad m_2 = 1 \]
Therefore, the slope of one of the lines is:
\[ m_1 = -1 \]

### Conclusion:
The slope of one of the lines is \(-1\).