If `lambdax^(2)-10xy+12y^(2)+5x-16y-3=0`, represents a pair of straight lines, then the value of `lambda` is

Options

A. 1
B. 2
C. `3//2`
D. 3

Explanation

To find the value of \(\lambda\) in the equation \( \lambda x^2 - 10xy + 12y^2 + 5x - 16y - 3 = 0 \) which represents a pair of straight lines, we can follow these steps:

### Step 1: Identify the coefficients
The general form of a pair of straight lines is given by:
\[ Ax^2 + Bxy + Cy^2 + 2Fx + 2Gy + D = 0 \]
From the given equation, we can identify the coefficients:
- \(A = \lambda\)
- \(B = -10\)
- \(C = 12\)
- \(F = \frac{5}{2}\)
- \(G = -8\)
- \(D = -3\)

### Step 2: Calculate the determinant
For the equation to represent a pair of straight lines, the determinant \(\Delta\) must be equal to zero:
\[ \Delta = \begin{vmatrix} A & \frac{B}{2} & F \\ \frac{B}{2} & C & G \\ F & G & D \end{vmatrix} \]
Substituting the values we identified:
\[ \Delta = \begin{vmatrix} \lambda & -5 & \frac{5}{2} \\ -5 & 12 & -8 \\ \frac{5}{2} & -8 & -3 \end{vmatrix} \]

### Step 3: Expand the determinant
Now we will calculate the determinant:
\[ \Delta = \lambda \begin{vmatrix} 12 & -8 \\ -8 & -3 \end{vmatrix} - (-5) \begin{vmatrix} -5 & -8 \\ \frac{5}{2} & -3 \end{vmatrix} + \frac{5}{2} \begin{vmatrix} -5 & 12 \\ \frac{5}{2} & -8 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \(\begin{vmatrix} 12 & -8 \\ -8 & -3 \end{vmatrix} = (12)(-3) - (-8)(-8) = -36 - 64 = -100\)
2. \(\begin{vmatrix} -5 & -8 \\ \frac{5}{2} & -3 \end{vmatrix} = (-5)(-3) - (-8)(\frac{5}{2}) = 15 + 40 = 55\)
3. \(\begin{vmatrix} -5 & 12 \\ \frac{5}{2} & -8 \end{vmatrix} = (-5)(-8) - (12)(\frac{5}{2}) = 40 - 30 = 10\)

Substituting back into the determinant:
\[ \Delta = \lambda(-100) + 5(55) + \frac{5}{2}(10) \]
\[ = -100\lambda + 275 + 25 = -100\lambda + 300 \]

### Step 4: Set the determinant to zero
To find \(\lambda\), we set the determinant equal to zero:
\[ -100\lambda + 300 = 0 \]

### Step 5: Solve for \(\lambda\)
Rearranging gives:
\[ 100\lambda = 300 \]
\[ \lambda = \frac{300}{100} = 3 \]

### Final Answer
Thus, the value of \(\lambda\) is:
\[ \lambda = 3 \]