Equation `(2 + lambda)x^2-2 lambdaxy+(lambda -1)y^2-4x-2=0` represents a hyperbola if

Options

A. `lambda=4`
B. `lambda=1`
C. `lambda=4//3`
D. `lambda=-1`

Answer

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

\[ (2 + \lambda)x^2 - 2\lambda a xy + (\lambda - 1)y^2 - 4x - 2 = 0 \]

represents a hyperbola, we need to analyze the coefficients of the quadratic equation in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

### Step 1: Identify Coefficients
From the given equation, we can identify the coefficients:
- \(A = 2 + \lambda\)
- \(B = -2\lambda\)
- \(C = \lambda - 1\)
- \(D = -4\)
- \(E = 0\)
- \(F = -2\)

### Step 2: Condition for Hyperbola
For the conic section to represent a hyperbola, the following condition must be satisfied:

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

### Step 3: Substitute Coefficients into the Condition
Substituting the identified coefficients into the condition:

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

### Step 4: Simplify the Inequality
Calculating \(B^2\):

\[ 4\lambda^2 \]

Calculating \(4AC\):

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

Now, substituting back into the inequality:

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

### Step 5: Further Simplification
Distributing the negative sign:

\[ 4\lambda^2 - 4\lambda^2 - 4\lambda + 8 > 0 \]

This simplifies to:

\[ -4\lambda + 8 > 0 \]

### Step 6: Solve the Inequality
Rearranging gives:

\[ 4\lambda < 8 \implies \lambda < 2 \]

### Step 7: Condition from the Determinant
Next, we also need to ensure that the determinant condition is satisfied. The determinant \(\Delta\) for the conic section is given by:

\[ \Delta = A \cdot C - \left(\frac{B}{2}\right)^2 \]

Calculating \(\Delta\):

\[ \Delta = (2 + \lambda)(\lambda - 1) - \left(-\lambda\right)^2 \]

Expanding this:

\[ = (2\lambda - 2 + \lambda^2 - \lambda) - \lambda^2 = 2\lambda - 2 - \lambda = \lambda - 2 \]

Setting the determinant condition:

\[ \lambda - 2 \neq 0 \implies \lambda \neq 2 \]

### Step 8: Combine Conditions
From the conditions derived:
1. \(\lambda < 2\)
2. \(\lambda \neq 2\)

Thus, the values of \(\lambda\) for which the equation represents a hyperbola are:

\[ \lambda < 2 \]

### Final Step: Conclusion
The equation represents a hyperbola for all values of \(\lambda\) such that:

\[ \lambda < 2 \]