If the equation `lambdax^2+ 4xy + y^2 + lambdax + 3y + 2 = 0` represent a parabola then find `lambda`.

Options

A. -4
B. 4
C. 0
D. none of these

Explanation

To determine the value of \( \lambda \) for which the equation

\[ \lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0 \]

represents a parabola, we will compare it to the general conic section equation:

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

### Step 1: Identify coefficients

From the given equation, we can identify the coefficients as follows:

- \( A = \lambda \)
- \( B = 4 \)
- \( C = 1 \)
- \( D = \lambda \)
- \( E = 3 \)
- \( F = 2 \)

### Step 2: Use the condition for a parabola

For the conic section to represent a parabola, the following condition must hold:

\[ B^2 - 4AC = 0 \]

### Step 3: Substitute the coefficients into the condition

Substituting the identified coefficients into the condition:

\[ 4^2 - 4(\lambda)(1) = 0 \]

### Step 4: Simplify the equation

Now, simplify the equation:

\[ 16 - 4\lambda = 0 \]

### Step 5: Solve for \( \lambda \)

Rearranging the equation gives:

\[ 4\lambda = 16 \]

Dividing both sides by 4:

\[ \lambda = 4 \]

### Conclusion

Thus, the value of \( \lambda \) for which the given equation represents a parabola is

\[ \lambda = 4 \]

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