If the equation `hxy + gx + fy + c = 0` represents a pair of straight lines, then

Options

A. `fg = ch`
B. `gh = fc`
C. `fh = gc`
D. `fh = -gc`

Explanation

To determine the relationship between the coefficients \( h \), \( g \), and \( f \) in the equation \( hxy + gx + fy + c = 0 \) that represents a pair of straight lines, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the General Form**:
The general form of the equation for a pair of straight lines is given by:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
In our case, we have \( a = 0 \), \( b = 0 \), \( 2h = h \), \( 2g = g \), \( 2f = f \), and \( c = c \).

2. **Use the Condition for Pair of Straight Lines**:
The condition for the equation to represent a pair of straight lines is:
\[ abc + 2fgh - a^2 - bg^2 - ch^2 = 0 \]
Substituting \( a = 0 \) and \( b = 0 \) into the equation gives:
\[ 0 + 2fgh - 0 - 0 - ch^2 = 0 \]
Simplifying this, we find:
\[ 2fgh - ch^2 = 0 \]

3. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 2fgh = ch^2 \]

4. **Dividing by \( h \) (assuming \( h \neq 0 \))**:
Dividing both sides by \( h \) (assuming \( h \neq 0 \)) results in:
\[ 2fg = ch \]

5. **Final Relationship**:
Thus, we can express the relationship as:
\[ fg = \frac{ch}{2} \]

### Conclusion:
The relationship between the coefficients \( h \), \( g \), and \( f \) in the equation \( hxy + gx + fy + c = 0 \) that represents a pair of straight lines is:
\[ fg = \frac{ch}{2} \]