If the equation `2x^(2)+2hxy +6y^(2) - 4x +5y -6 = 0` represents a pair of straight lines, then the length of intercept on the x-axis cut by the lines is equal to

Options

A. 2
B. 4
C. `sqrt(7)`
D. 0

Explanation

To solve the problem, we need to determine the length of the intercept on the x-axis cut by the lines represented by the given equation:

**Given equation:**
\[ 2x^2 + 2hxy + 6y^2 - 4x + 5y - 6 = 0 \]

### Step 1: Identify coefficients
We compare the given equation with the general form of the equation of a pair of straight lines:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

From the given equation, we can identify the coefficients:
- \( a = 2 \)
- \( h = h \) (remains as is since it's not specified)
- \( b = 6 \)
- \( g = -2 \) (from the term \(-4x\))
- \( f = \frac{5}{2} \) (from the term \(5y\))
- \( c = -6 \)

### Step 2: Use the formula for x-intercept
The formula for the length of the x-intercept cut by the lines is given by:
\[ \text{Length of x-intercept} = \frac{2\sqrt{g^2 - ac}}{|a|} \]

### Step 3: Substitute the values into the formula
We need to calculate \( g^2 - ac \):
- \( g = -2 \)
- \( a = 2 \)
- \( c = -6 \)

Calculating \( g^2 \):
\[ g^2 = (-2)^2 = 4 \]

Calculating \( ac \):
\[ ac = 2 \times (-6) = -12 \]

Now substituting into the formula:
\[ g^2 - ac = 4 - (-12) = 4 + 12 = 16 \]

### Step 4: Calculate the length of the x-intercept
Now substituting back into the formula:
\[ \text{Length of x-intercept} = \frac{2\sqrt{16}}{|2|} \]
\[ = \frac{2 \times 4}{2} = \frac{8}{2} = 4 \]

### Final Answer
The length of the intercept on the x-axis cut by the lines is:
\[ \boxed{4} \]