The equation `ax^(2)+by^(2)+lamdax+lamday=0,(lamda!=0)` represents a pair of straight lines if

Options

A. `b+c=0`
B. `c+a=0`
C. `a+b=0`
D. none of these

Answer

To determine the conditions under which the equation \( ax^2 + by^2 + \lambda x + \lambda y = 0 \) (where \( \lambda \neq 0 \)) represents a pair of straight lines, we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be rewritten in the standard form for a conic section:
\[ Ax^2 + By^2 + 2Gxy + 2Fx + 2Hy + C = 0 \]
In our case:
- \( A = a \)
- \( B = b \)
- \( G = 0 \) (since there is no \( xy \) term)
- \( F = \frac{\lambda}{2} \)
- \( H = \frac{\lambda}{2} \)
- \( C = 0 \)

### Step 2: Write the discriminant condition
For a conic section to represent a pair of straight lines, the discriminant \( D \) must be equal to zero:
\[ D = ABC + 2GFH - AF^2 - BG^2 - CH^2 \]
Substituting the values we identified:
\[ D = a \cdot b \cdot 0 + 2 \cdot 0 \cdot \frac{\lambda}{2} \cdot \frac{\lambda}{2} - a \left(\frac{\lambda}{2}\right)^2 - b \cdot 0^2 - 0 \cdot \left(\frac{\lambda}{2}\right)^2 \]
This simplifies to:
\[ D = -a \left(\frac{\lambda^2}{4}\right) = -\frac{a \lambda^2}{4} \]

### Step 3: Set the discriminant to zero
For the equation to represent a pair of straight lines, we set the discriminant \( D \) to zero:
\[ -\frac{a \lambda^2}{4} = 0 \]
Since \( \lambda \neq 0 \), we conclude that:
\[ a = 0 \]

### Step 4: Analyze the condition for \( b \)
Now, we also need to consider the condition for \( b \). Since we are looking for a pair of straight lines, we also require:
\[ b = 0 \]
Thus, we find that:
\[ a + b = 0 \]

### Conclusion
The conditions under which the equation \( ax^2 + by^2 + \lambda x + \lambda y = 0 \) represents a pair of straight lines are:
\[ a + b = 0 \quad \text{and} \quad \lambda \neq 0 \]