The equation `x^(3)+ax^(2)y+bxy^(2)+y^(3)=0` represents three straight lines, two of which are perpendicular, then the equation of the third line, is

Options

A. `y=ax`
B. `y=bx`
C. `y=x`
D. `y=-x`

Explanation

To solve the problem, we need to find the equation of the third line given that the equation \(x^3 + ax^2y + bxy^2 + y^3 = 0\) represents three straight lines, two of which are perpendicular.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The given equation can be factored into three linear factors since it represents three straight lines. We can express these lines in the form:
\[ y = m_1x, \quad y = m_2x, \quad y = m_3x \]
where \(m_1\), \(m_2\), and \(m_3\) are the slopes of the three lines.

**Hint**: Remember that if two lines are perpendicular, the product of their slopes is \(-1\).

2. **Identifying Perpendicular Lines**:
Since two of the lines are perpendicular, we can assume:
\[ m_1 \cdot m_2 = -1 \]
This implies that if we let \(m_1 = m\), then \(m_2 = -\frac{1}{m}\).

**Hint**: Use the relationship of slopes for perpendicular lines.

3. **Finding the Third Slope**:
The product of the slopes of all three lines can be expressed as:
\[ m_1 \cdot m_2 \cdot m_3 = -1 \]
Substituting \(m_1\) and \(m_2\):
\[ m \cdot \left(-\frac{1}{m}\right) \cdot m_3 = -1 \]
Simplifying this gives:
\[ -1 \cdot m_3 = -1 \implies m_3 = 1 \]

**Hint**: Remember to isolate \(m_3\) after substituting the known values.

4. **Writing the Equation of the Third Line**:
Since \(m_3 = 1\), the equation of the third line can be written as:
\[ y = m_3x = x \]

**Hint**: The equation of a line through the origin with slope \(m\) is \(y = mx\).

5. **Conclusion**:
Therefore, the equation of the third line is:
\[ y = x \]

### Final Answer:
The equation of the third line is \(y = x\).