Let `(a,b,c)ne(0,0,0)`. The pair of equations which does not represent a straight line is

Options

A. `ax-by+cz+d=0,ax+b'y+cz+d=0(bneb')`
B. `ax-by+cz+d=0,ax+by+c'z+d=0(cnec')`
C. `ax+by+cz+d=0,ax+by+cz-d'=0(dned')`
D. `ax+by+cz+d=0,a'x+by+cz+d=0(anea')`

Answer

To determine which pair of equations does not represent a straight line, we need to analyze the given equations of two planes. The general form of the equations of two planes can be written as:

1. \( A_1x + B_1y + C_1z + D_1 = 0 \) (Plane 1)
2. \( A_2x + B_2y + C_2z + D_2 = 0 \) (Plane 2)

### Step 1: Identify the conditions for the intersection of two planes
Two planes will intersect along a straight line unless they are parallel. For two planes to be parallel, the following conditions must hold:

- The ratios of the coefficients of \(x\), \(y\), and \(z\) must be equal:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

### Step 2: Analyze the options
We need to check the given options to see which pair of equations satisfies the parallel condition.

1. **Option 1**: \( A_1x + B_1y + C_1z + D_1 = 0 \) and \( A_2x + B_2y + C_2z + D_2 = 0 \)
- Check if \( \frac{A_1}{A_2} \), \( \frac{B_1}{B_2} \), and \( \frac{C_1}{C_2} \) are equal.
- If not equal, they intersect in a line.

2. **Option 2**: Similar analysis as Option 1.

3. **Option 3**: Check if \( \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \).
- If all ratios are equal, the planes are parallel and do not intersect in a line.

4. **Option 4**: Check if they intersect or are parallel.

### Step 3: Determine the option that does not represent a straight line
From the analysis, if we find that one of the options satisfies the condition for parallel planes (i.e., all ratios are equal), that option will be the one that does not represent a straight line.

### Conclusion
After checking all options, we conclude that **Option 3** does not represent a straight line because the planes are parallel.