If the equation
`ax^(2)+2hxy+by^(2)+2gx+2fy+c=0`
represents two straights lines, then the product of the perpendicular from the origin on these straight lines, is

Options

A. `(c)/(sqrt((a-b)^(2)-4h^(2)))`
B. `(c)/(sqrt((a-b)^(2)+4h^(2)))`
C. `(c)/(sqrt((a+b)^(2)-4h^(2)))`
D. none of these

Explanation

To solve the problem, we need to find the product of the perpendicular distances from the origin to the two straight lines represented by the given equation:

\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

### Step-by-Step Solution:

1. **Understand the Equation**:
The given equation represents a pair of straight lines if the determinant formed by the coefficients is zero. The general form of the equation is:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

2. **Identify the Lines**:
We can express the equation in terms of two lines \( L_1 \) and \( L_2 \):
\[ L_1: l_1 x + m_1 y + n_1 = 0 \]
\[ L_2: l_2 x + m_2 y + n_2 = 0 \]

3. **Coefficients Comparison**:
By comparing coefficients of the original equation with the expanded form of the product of the two lines, we have:
- \( l_1 l_2 = a \)
- \( m_1 m_2 = b \)
- \( n_1 n_2 = c \)
- \( l_1 m_2 + l_2 m_1 = 2h \)
- \( l_1 n_2 + l_2 n_1 = 2g \)
- \( m_1 n_2 + m_2 n_1 = 2f \)

4. **Formula for Product of Perpendicular Distances**:
The product of the perpendicular distances from the origin to the two lines can be calculated using the formula:
\[ \text{Product of perpendiculars} = \frac{n_1 n_2}{\sqrt{(l_1^2 + m_1^2)(l_2^2 + m_2^2)}} \]

5. **Substituting Values**:
From the coefficients, we know:
- \( n_1 n_2 = c \)
- \( l_1 l_2 = a \)
- \( m_1 m_2 = b \)
- \( l_1 m_2 + l_2 m_1 = 2h \)

Therefore, we can express the denominator:
\[ \sqrt{(l_1^2 + m_1^2)(l_2^2 + m_2^2)} = \sqrt{l_1^2 l_2^2 + m_1^2 m_2^2 + (l_1 m_2 + l_2 m_1)^2 - 2l_1 l_2 m_1 m_2} \]
Substituting the known values:
\[ = \sqrt{a^2 + b^2 + 4h^2 - 2ab} \]

6. **Final Expression**:
Thus, the product of the perpendicular distances from the origin to the two lines is:
\[ \frac{c}{\sqrt{(a - b)^2 + 4h^2}} \]

### Conclusion:
The product of the perpendicular distances from the origin to the two straight lines represented by the equation is given by:
\[ \frac{c}{\sqrt{(a - b)^2 + 4h^2}} \]