If the angle between the straight lines represented by `2x^(2)+5xy+3y^(2)+7y+4=0` is `tan^(-1)m` , then `[(1)/(pi m)]` is (where `[.]` denote greatest integer function) (a) 0 (b) 1 (c) 2 (d) 3

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

To solve the problem, we need to find the angle between the straight lines represented by the equation \(2x^2 + 5xy + 3y^2 + 7y + 4 = 0\) and then calculate \(\left\lfloor \frac{1}{\pi m} \right\rfloor\), where \(m\) is related to the angle.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is \(2x^2 + 5xy + 3y^2 + 7y + 4 = 0\). We can identify the coefficients as:
- \(a = 2\)
- \(b = 3\)
- \(h = \frac{5}{2}\) (since \(2h = 5\))

2. **Use the formula for the angle between two lines**:
The angle \(\theta\) between the two lines represented by the equation is given by:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Substituting the values:
\[ h^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]
\[ ab = 2 \cdot 3 = 6 \]
Therefore,
\[ h^2 - ab = \frac{25}{4} - 6 = \frac{25}{4} - \frac{24}{4} = \frac{1}{4} \]

3. **Calculate \(\tan \theta\)**:
Now substituting these values into the formula:
\[ \tan \theta = \frac{2\sqrt{\frac{1}{4}}}{2 + 3} = \frac{2 \cdot \frac{1}{2}}{5} = \frac{1}{5} \]
Since we are given that \(\theta = \tan^{-1} m\), we have:
\[ m = \frac{1}{5} \]

4. **Calculate \(\left\lfloor \frac{1}{\pi m} \right\rfloor\)**:
Now we need to calculate:
\[ \frac{1}{\pi m} = \frac{1}{\pi \cdot \frac{1}{5}} = \frac{5}{\pi} \]
Using \(\pi \approx 3.14\):
\[ \frac{5}{\pi} \approx \frac{5}{3.14} \approx 1.59 \]
Therefore, the greatest integer function gives us:
\[ \left\lfloor 1.59 \right\rfloor = 1 \]

5. **Final Answer**:
Thus, the answer is:
\[ \boxed{1} \]