The condition that one of the straight lines given by the equation `ax^(2)+2hxy+by^(2)=0` may coincide with one of those given by the equation `a'x^(2)+2h'xy+b'y^(2)=0` is

Options

A. `(ab'-a'b)^(2)=4(ha'-h'a)(bh'-b'h)`
B. `(ab'=a'b)^(2)=(ha'-h'a)(bh'-b'h)`
C. `(ha'-h'a)=4(ab'-a'b)(bh'-b'h)`
D. `(bh'-b'h)^(2)=4(ab'-a'b)(ha'-h'a)`

Explanation

To find the condition that one of the straight lines given by the equation \( ax^2 + 2hxy + by^2 = 0 \) may coincide with one of those given by the equation \( a'x^2 + 2h'xy + b'y^2 = 0 \), we can follow these steps:

### Step-by-Step Solution

1. **Assume the Common Line**:
Let the common line be represented by the equation \( y = mx \), where \( m \) is the slope of the line.

2. **Substitute into the First Equation**:
Substitute \( y = mx \) into the first equation:
\[ a x^2 + 2h (mx)x + b (mx)^2 = 0 \]
This simplifies to:
\[ (b m^2 + 2hm + a)x^2 = 0 \]
For this equation to hold for all \( x \), the coefficient must equal zero:
\[ b m^2 + 2hm + a = 0 \tag{1} \]

3. **Substitute into the Second Equation**:
Similarly, substitute \( y = mx \) into the second equation:
\[ a' x^2 + 2h' (mx)x + b' (mx)^2 = 0 \]
This simplifies to:
\[ (b' m^2 + 2h'm + a')x^2 = 0 \]
Again, for this to hold for all \( x \), the coefficient must equal zero:
\[ b' m^2 + 2h'm + a' = 0 \tag{2} \]

4. **Set Up the System of Equations**:
Now we have two equations (1) and (2):
\[ b m^2 + 2hm + a = 0 \tag{1} \]
\[ b' m^2 + 2h'm + a' = 0 \tag{2} \]

5. **Eliminate \( m \)**:
To eliminate \( m \), we can solve these equations simultaneously. We can express \( m \) in terms of \( a, b, h \) from equation (1) and substitute it into equation (2).

Rearranging equation (1):
\[ m = \frac{-b m^2 - a}{2h} \tag{3} \]

Substitute \( m \) from equation (3) into equation (2) and simplify to find a relationship between the coefficients \( a, b, h, a', b', h' \).

6. **Final Condition**:
After simplification, we arrive at the condition:
\[ (ab' - a'b)^2 = 4(h a' - h' a)(b h' - b' h) \]

### Conclusion
The condition that one of the straight lines given by the first equation may coincide with one of those given by the second equation is:
\[ (ab' - a'b)^2 = 4(h a' - h' a)(b h' - b' h) \]