The set of value of h for which the equation `4x^(2)+hxy-3y^(2)=0` represents a pair of real and distinct lines is

Options

A. R
B. `(3, 4)`
C. `(-3,4)`
D. `(4,oo)`

Explanation

To determine the set of values of \( h \) for which the equation \( 4x^2 + hxy - 3y^2 = 0 \) represents a pair of real and distinct lines, we follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared with the general form of a conic section:
\[ Ax^2 + Bxy + Cy^2 = 0 \]
Here, we have:
- \( A = 4 \)
- \( B = h \)
- \( C = -3 \)

### Step 2: Apply the condition for real and distinct lines
For the equation to represent a pair of real and distinct lines, the following condition must be satisfied:
\[ B^2 - 4AC > 0 \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ h^2 - 4(4)(-3) > 0 \]

### Step 3: Simplify the inequality
Calculating \( 4(4)(-3) \):
\[ h^2 - 4 \times 4 \times -3 = h^2 + 48 > 0 \]

### Step 4: Analyze the inequality
Since \( h^2 \) is always non-negative (as it is a square), the expression \( h^2 + 48 \) is always positive for any real value of \( h \). Therefore:
\[ h^2 + 48 > 0 \quad \text{is always true.} \]

### Conclusion
The equation \( 4x^2 + hxy - 3y^2 = 0 \) represents a pair of real and distinct lines for all values of \( h \).

### Final Answer
The set of values of \( h \) is:
\[ \text{All real numbers } h \in \mathbb{R} \]

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