The equation `3x^2+2hxy+3y^2=0` represents a pair of straight lines passing through the origin . The two lines are

Options

A. real and distinct , if `h^2gt3`
B. real and distinct , if` h^2 gt 9`
C. real and coincident , if `h^2 =3`
D. real and coincident , if `h^2gt 3`

Answer

To solve the problem, we need to analyze the given equation \(3x^2 + 2hxy + 3y^2 = 0\) and determine the conditions under which it represents a pair of straight lines passing through the origin.

### Step-by-Step Solution:

1. **Identify the general form of the equation**:
The given equation is of the form \(Ax^2 + Bxy + Cy^2 = 0\), where:
- \(A = 3\)
- \(B = 2h\)
- \(C = 3\)

2. **Apply the condition for a pair of straight lines**:
For the equation to represent a pair of straight lines, the condition is:
\[ B^2 - 4AC > 0 \]
Substituting the values of \(A\), \(B\), and \(C\):
\[ (2h)^2 - 4(3)(3) > 0 \]

3. **Simplify the inequality**:
\[ 4h^2 - 36 > 0 \]
This simplifies to:
\[ 4h^2 > 36 \]

4. **Divide by 4**:
\[ h^2 > 9 \]

5. **Take the square root**:
This gives us:
\[ |h| > 3 \]
Therefore, \(h > 3\) or \(h < -3\).

6. **Conclusion**:
Since \(h^2 > 9\) indicates that the lines are real and distinct, we conclude that the equation \(3x^2 + 2hxy + 3y^2 = 0\) represents a pair of straight lines passing through the origin when \(h\) satisfies the condition \(h > 3\) or \(h < -3\).

### Final Answer:
The two lines represented by the equation are real and distinct when \(h^2 > 9\).

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