If two of the three lines represented by `ax^3+ bx^2y +cxy^2+dy^3=0` may be at right angles then

Answer

To solve the problem, we need to analyze the given cubic equation representing three lines and determine the condition under which two of these lines are perpendicular.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The given equation is:
\[ ax^3 + bx^2y + cxy^2 + dy^3 = 0 \]
This represents three lines in the plane.

2. **Condition for Perpendicular Lines**:
For two lines to be perpendicular, the sum of the coefficients of \(x^2\) and \(y^2\) in the equation must equal zero. This is derived from the general property of perpendicular lines.

3. **Identifying Coefficients**:
In our equation:
- The coefficient of \(x^2\) is \(b\).
- The coefficient of \(y^2\) is \(c\).

Therefore, the condition for the lines to be perpendicular can be expressed as:
\[ b + c = 0 \quad \text{(Equation 1)} \]

4. **Expressing the Equation of Perpendicular Lines**:
We can represent the pair of perpendicular lines in the form:
\[ x^2 + pxy - y^2 = 0 \]
To relate this to our cubic equation, we need to multiply it by a linear factor, say \(Ax - Dy\).

5. **Expanding the Product**:
Multiplying the equation of perpendicular lines by \(Ax - Dy\):
\[ (x^2 + pxy - y^2)(Ax - Dy) \]
Expanding this, we get:
\[ Ax^3 + pAx^2y - Ay^2x - Dy^2x + pDxy^2 - Dy^3 \]
Rearranging gives:
\[ Ax^3 + (pA - D)x^2y + (pD - A)xy^2 - Dy^3 = 0 \]

6. **Equating Coefficients**:
Now, we equate the coefficients from both sides of the original equation and the expanded equation:
- Coefficient of \(x^2y\):
\[ b = pA - D \quad \text{(Equation 2)} \]
- Coefficient of \(xy^2\):
\[ c = pD - A \quad \text{(Equation 3)} \]

7. **Solving for \(p\)**:
From Equation 2:
\[ pA = b + D \implies p = \frac{b + D}{A} \]
From Equation 3:
\[ pD = c + A \implies p = \frac{c + A}{D} \]

8. **Equating the Two Expressions for \(p\)**:
Setting the two expressions for \(p\) equal gives:
\[ \frac{b + D}{A} = \frac{c + A}{D} \]

9. **Cross Multiplying**:
Cross multiplying yields:
\[ (b + D)D = (c + A)A \]
Expanding this gives:
\[ bD + D^2 = cA + A^2 \]

10. **Rearranging**:
Rearranging the equation leads to:
\[ A^2 + bD + cA + D^2 = 0 \]

### Final Condition:
Thus, the condition for two of the three lines represented by the cubic equation to be at right angles is:
\[ A^2 + bD + cA + D^2 = 0 \]