If the equation `x^2+(lambda+mu)x y+lambdau y^2+x+muy=0` represents two parallel straight lines, then prove that `lambda=mudot`

Options

A. (3,-1)
B. `-3,1`
C. (1,1)
D. none of these

Answer

To prove that \( \lambda = \mu \) given the equation

\[ x^2 + (\lambda + \mu)xy + \lambda \mu y^2 + x + \mu y = 0 \]

represents two parallel straight lines, we will follow these steps:

### Step 1: Understanding the Condition for Parallel Lines
For the equation to represent two parallel lines, the slopes of these lines must be equal. This means that the coefficient of \( xy \) must be such that it allows for two lines with the same slope.

### Step 2: Identifying the Coefficients
The given equation can be rewritten in the standard form of a conic section. The general form for two lines is:

\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

Here, we identify:
- \( A = 1 \)
- \( B = \lambda + \mu \)
- \( C = \lambda \mu \)
- \( D = 1 \)
- \( E = \mu \)
- \( F = 0 \)

### Step 3: Condition for Parallel Lines
For the equation to represent two parallel lines, the discriminant must be zero. The discriminant \( D \) for the conic section is given by:

\[ D = B^2 - 4AC \]

Substituting our values:

\[ D = (\lambda + \mu)^2 - 4(1)(\lambda \mu) \]

### Step 4: Setting the Discriminant to Zero
Setting the discriminant to zero for parallel lines:

\[ (\lambda + \mu)^2 - 4\lambda \mu = 0 \]

### Step 5: Expanding and Rearranging
Expanding the equation:

\[ \lambda^2 + 2\lambda\mu + \mu^2 - 4\lambda\mu = 0 \]

This simplifies to:

\[ \lambda^2 - 2\lambda\mu + \mu^2 = 0 \]

### Step 6: Factoring the Quadratic
This can be factored as:

\[ (\lambda - \mu)^2 = 0 \]

### Step 7: Conclusion
From the factored form, we conclude that:

\[ \lambda - \mu = 0 \implies \lambda = \mu \]

Thus, we have proved that if the given equation represents two parallel straight lines, then \( \lambda = \mu \).

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