If the quadrilateral formed by the lines
`ax+by+c=0,a'x+b'y+c=0`
`ax+by+c'=0,a'x+b'y+c'=0`
have perpendicular diagonals then

Options

A. `b^(2)+c^(2)=b^('2)+c^('2)`
B. `c^(2)+a^(2)=c^('2)+a^('2)`
C. `a^(2)+b^(2)=a^('2)+b^('2)`
D. None of these

Answer

To solve the problem, we need to analyze the conditions under which the diagonals of the quadrilateral formed by the given lines are perpendicular.

### Step-by-Step Solution:

1. **Identify the Lines:**
The given lines are:
- Line 1: \( ax + by + c = 0 \)
- Line 2: \( a'x + b'y + c = 0 \)
- Line 3: \( ax + by + c' = 0 \)
- Line 4: \( a'x + b'y + c' = 0 \)

2. **Understanding the Quadrilateral:**
The quadrilateral is formed by the intersection points of these lines. The lines \( ax + by + c = 0 \) and \( ax + by + c' = 0 \) are parallel, and the lines \( a'x + b'y + c = 0 \) and \( a'x + b'y + c' = 0 \) are also parallel.

3. **Finding the Intersection Points:**
The intersection points of these lines can be found by solving pairs of equations:
- Intersection of Line 1 and Line 2
- Intersection of Line 1 and Line 4
- Intersection of Line 3 and Line 2
- Intersection of Line 3 and Line 4

4. **Condition for Perpendicular Diagonals:**
For the diagonals of the quadrilateral to be perpendicular, the slopes of the diagonals must satisfy the condition:
\[ m_1 \cdot m_2 = -1 \]
where \( m_1 \) and \( m_2 \) are the slopes of the diagonals.

5. **Finding the Slopes:**
The slopes of the lines can be calculated as follows:
- Slope of Line 1: \( m_1 = -\frac{a}{b} \)
- Slope of Line 2: \( m_2 = -\frac{a'}{b'} \)
- Slope of Line 3: \( m_3 = -\frac{a}{b} \)
- Slope of Line 4: \( m_4 = -\frac{a'}{b'} \)

6. **Using the Distance Formula:**
The distance between two parallel lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by:
\[ d = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \]
Thus, the distances between the pairs of lines can be calculated.

7. **Establishing the Relationship:**
Since the diagonals are perpendicular, we can derive the relationship:
\[ a^2 + b^2 = a'^2 + b'^2 \]

### Conclusion:
The condition for the quadrilateral formed by the given lines to have perpendicular diagonals is:
\[ a^2 + b^2 = a'^2 + b'^2 \]