`6x^(2)+hxy+12y^(2)=0` represents pair of parallel straight lines, if `h` is

Options

A. `+-6sqrt(2)`
B. `+-sqrt(2)`
C. `+-12sqrt(2)`
D. `+-sqrt(6)`

Explanation

To determine the value of \( h \) such that the equation \( 6x^2 + hxy + 12y^2 = 0 \) represents a pair of parallel straight lines, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is in the form \( ax^2 + hxy + by^2 = 0 \).
Here, \( a = 6 \), \( b = 12 \), and \( h = h \).

2. **Use the condition for parallel lines**:
For the pair of straight lines to be parallel, the condition we need to satisfy is:
\[ h^2 - 4ab = 0 \]
where \( a = 6 \) and \( b = 12 \).

3. **Substitute the values of \( a \) and \( b \)**:
Substitute \( a \) and \( b \) into the condition:
\[ h^2 - 4(6)(12) = 0 \]

4. **Calculate \( 4ab \)**:
Calculate \( 4ab \):
\[ 4 \cdot 6 \cdot 12 = 288 \]
Therefore, the equation becomes:
\[ h^2 - 288 = 0 \]

5. **Solve for \( h^2 \)**:
Rearranging gives:
\[ h^2 = 288 \]

6. **Find \( h \)**:
Taking the square root of both sides:
\[ h = \pm \sqrt{288} \]

7. **Simplify \( \sqrt{288} \)**:
We can simplify \( \sqrt{288} \):
\[ \sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2} \]
Thus, we have:
\[ h = \pm 12\sqrt{2} \]

### Final Answer:
The values of \( h \) such that the equation represents a pair of parallel straight lines are:
\[ h = 12\sqrt{2} \quad \text{or} \quad h = -12\sqrt{2} \]

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