The value of `h` for which `3x^2-2hxy + 4y^2 = 0 `represents a pair of coincident lines are

Options

A. `+-3sqrt3`
B. `+-sqrt3`
C. `+-2sqrt3`
D. `+-sqrt6`

Answer

To find the value of \( h \) for which the equation \( 3x^2 - 2hxy + 4y^2 = 0 \) represents a pair of coincident lines, we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared with the general form of a conic section:
\[ ax^2 + 2hxy + by^2 = 0 \]
From the equation \( 3x^2 - 2hxy + 4y^2 = 0 \), we can identify:
- \( a = 3 \)
- \( b = 4 \)
- \( 2h = -2h \)

### Step 2: Use the condition for coincident lines
For the equation to represent a pair of coincident lines, the condition is:
\[ h^2 - ab = 0 \]
Substituting the values of \( a \) and \( b \):
\[ h^2 - (3)(4) = 0 \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ h^2 - 12 = 0 \]

### Step 4: Solve for \( h \)
To find \( h \), we can rearrange the equation:
\[ h^2 = 12 \]
Taking the square root of both sides gives:
\[ h = \pm \sqrt{12} \]
This can be simplified further:
\[ h = \pm 2\sqrt{3} \]

### Final Answer
Thus, the values of \( h \) for which the equation represents a pair of coincident lines are:
\[ h = 2\sqrt{3} \quad \text{and} \quad h = -2\sqrt{3} \]

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