Consider the equation of a pair of straight lines as `x^2-3xy+lambday^2+3x=5y+2=0`
The value of `lambda` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \(\lambda\) in the equation of a pair of straight lines given by:

\[ x^2 - 3xy + \lambda y^2 + 3x - 5y + 2 = 0 \]

we can follow these steps:

### Step 1: Identify the coefficients
The general form of the equation of a pair of straight lines is given by:

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

From the given equation, we can identify the coefficients:
- \(A = 1\)
- \(B = -3\)
- \(C = \lambda\)
- \(D = 3\)
- \(E = -5\)
- \(F = 2\)

### Step 2: Set up the determinant condition
For the equation to represent a pair of straight lines, the determinant of the matrix formed by the coefficients must be zero:

\[ \begin{vmatrix} A & \frac{B}{2} & \frac{D}{2} \\ \frac{B}{2} & C & \frac{E}{2} \\ \frac{D}{2} & \frac{E}{2} & F \end{vmatrix} = 0 \]

Substituting the values we identified, we have:

\[ \begin{vmatrix} 1 & -\frac{3}{2} & \frac{3}{2} \\ -\frac{3}{2} & \lambda & -\frac{5}{2} \\ \frac{3}{2} & -\frac{5}{2} & 2 \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
Now, we calculate the determinant:

\[ = 1 \left( \lambda \cdot 2 - \left(-\frac{5}{2}\right) \cdot \left(-\frac{5}{2}\right) \right) - \left(-\frac{3}{2}\right) \left(-\frac{3}{2} \cdot 2 - \frac{3}{2} \cdot -\frac{5}{2}\right) + \frac{3}{2} \left(-\frac{3}{2} \cdot -\frac{5}{2} - \lambda \cdot \frac{3}{2}\right) \]

Calculating each term:

1. First term:
\[ 1 \left( 2\lambda - \frac{25}{4} \right) = 2\lambda - \frac{25}{4} \]

2. Second term:
\[ -\left(-\frac{3}{2}\right) \left(-\frac{3}{2} \cdot 2 + \frac{15}{4}\right) = \frac{3}{2} \left(-3 + \frac{15}{4}\right) = \frac{3}{2} \left(-\frac{12}{4} + \frac{15}{4}\right) = \frac{3}{2} \cdot \frac{3}{4} = \frac{9}{8} \]

3. Third term:
\[ \frac{3}{2} \left(\frac{15}{4} - \frac{3\lambda}{2}\right) = \frac{3}{2} \cdot \frac{15}{4} - \frac{9\lambda}{4} = \frac{45}{8} - \frac{9\lambda}{4} \]

### Step 4: Combine and set equal to zero
Combining all the terms gives us:

\[ 2\lambda - \frac{25}{4} - \frac{9}{8} + \frac{45}{8} - \frac{9\lambda}{4} = 0 \]

Simplifying this:

\[ 2\lambda - \frac{25}{4} + \frac{36}{8} - \frac{9\lambda}{4} = 0 \]

Converting everything to a common denominator (which is 8):

\[ 2\lambda - \frac{50}{8} + \frac{36}{8} - \frac{18\lambda}{8} = 0 \]

Combining like terms:

\[ \left(16\lambda - 18\lambda\right)/8 - \frac{14}{8} = 0 \]

This simplifies to:

\[ -2\lambda - \frac{14}{8} = 0 \]

### Step 5: Solve for \(\lambda\)
Now, solving for \(\lambda\):

\[ -2\lambda = \frac{14}{8} \]
\[ \lambda = -\frac{14}{16} = -\frac{7}{8} \]

However, we made an error in the simplification. Let's go back to the determinant condition:

\[ 2\lambda - \frac{25}{4} + \frac{9}{8} + \frac{45}{8} - \frac{9\lambda}{4} = 0 \]

This leads to:

\[ 2\lambda - \frac{25}{4} + \frac{54}{8} - \frac{9\lambda}{4} = 0 \]

Combining terms gives:

\[ (2 - \frac{9}{4})\lambda = \frac{25}{4} - \frac{54}{8} \]

Solving this will yield the correct value of \(\lambda\).

### Final Answer
After solving the equation correctly, we find that the value of \(\lambda\) is:

\[ \lambda = 2 \]