Select and write the most appropriate answer from the given alternatives in each of the following sub-questions :
The slopes of the lines given by `12x^(2)+bxy-y^(2)=0` differ by 7. Then the value of b is :

Options

A. 2
B. `+-2`
C. `+-1`
D. 1

Explanation

To solve the problem, we need to find the value of \( b \) such that the slopes of the lines represented by the equation \( 12x^2 + bxy - y^2 = 0 \) differ by 7.

### Step 1: Identify the equation of the pair of lines
The given equation is \( 12x^2 + bxy - y^2 = 0 \). This represents a pair of straight lines passing through the origin.

### Step 2: Use the formula for the slopes of the lines
For a general quadratic equation of the form \( Ax^2 + Bxy + Cy^2 = 0 \), the slopes \( m_1 \) and \( m_2 \) of the lines can be found using the formula:
\[ m_{1,2} = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
In our case, \( A = 12 \), \( B = b \), and \( C = -1 \).

### Step 3: Calculate the slopes
Substituting the values into the formula:
\[ m_{1,2} = \frac{-b \pm \sqrt{b^2 - 4 \cdot 12 \cdot (-1)}}{2 \cdot 12} \]
This simplifies to:
\[ m_{1,2} = \frac{-b \pm \sqrt{b^2 + 48}}{24} \]

### Step 4: Find the difference of the slopes
The difference of the slopes is given as:
\[ |m_1 - m_2| = 7 \]
Using the formula for the difference of the slopes:
\[ |m_1 - m_2| = \frac{\sqrt{b^2 + 48}}{12} \]
Setting this equal to 7:
\[ \frac{\sqrt{b^2 + 48}}{12} = 7 \]

### Step 5: Solve for \( b \)
Multiplying both sides by 12:
\[ \sqrt{b^2 + 48} = 84 \]
Squaring both sides:
\[ b^2 + 48 = 7056 \]
Subtracting 48 from both sides:
\[ b^2 = 7008 \]
Taking the square root:
\[ b = \pm \sqrt{7008} \]

### Step 6: Simplify \( \sqrt{7008} \)
Calculating \( \sqrt{7008} \):
\[ 7008 = 16 \times 438 \]
Thus,
\[ b = \pm 4\sqrt{438} \]

### Conclusion
The value of \( b \) is \( \pm 4\sqrt{438} \).