If `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0` represents two parallel straight lines, then

Options

A. `h^(2)=ab`
B. `bg^(2)=af^(2)`
C. `hg=af`
D. all of these

Explanation

To determine the conditions under which the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) represents two parallel straight lines, we can follow these steps:

### Step 1: Understand the Form of the Equation
The given equation is a second-degree polynomial in \( x \) and \( y \). For it to represent two straight lines, it must be factorable into the product of two linear equations.

### Step 2: Factorization into Linear Equations
Assume the two parallel lines can be expressed as:
\[ (lx + my + n_1)(lx + my + n_2) = 0 \]
When expanded, this gives:
\[ l^2x^2 + 2lmxy + m^2y^2 + (ln_1 + ln_2)xy + (mn_1 + mn_2)y + n_1n_2 = 0 \]

### Step 3: Compare Coefficients
By comparing coefficients from the original equation and the expanded form, we have:
1. \( l^2 = a \)
2. \( 2lm = 2h \) → \( lm = h \)
3. \( m^2 = b \)
4. \( ln_1 + ln_2 = 2g \)
5. \( mn_1 + mn_2 = 2f \)
6. \( n_1n_2 = c \)

### Step 4: Conditions for Parallel Lines
For the lines to be parallel, the slopes of the lines must be equal. The slope of the lines can be derived from the coefficients \( l \) and \( m \). The condition for parallel lines is that the ratio of the coefficients of \( x \) and \( y \) should be the same, i.e., \( \frac{l}{m} \) is constant.

### Step 5: Derive Relationships
From the relationships derived:
- From \( lm = h \), we can express \( h^2 = l^2m^2 = ab \).
- From \( \frac{g}{f} = \frac{l}{m} \), we can derive \( bg^2 = af^2 \).
- From \( hg = af \), we can derive \( h^2g^2 = a^2f^2 \).

### Step 6: Conclusion
Thus, we have the following conditions:
1. \( h^2 = ab \)
2. \( bg^2 = af^2 \)
3. \( hg = af \)

Therefore, all three conditions must hold for the given quadratic equation to represent two parallel straight lines.

### Final Answer
The conditions are:
- \( h^2 = ab \)
- \( bg^2 = af^2 \)
- \( hg = af \)