If the quadrilateral formed by the lines `a x+b y+c=0,a^(prime)x+b^(prime)y+c=0,a x+b y+c^(prime)=0,a^(prime)x+b^(prime)y+c^(prime)=0` has perpendicular diagonals, then `b^2+c^2=b^('2)+c^('2)` `c^2+a^2=c^('2)+a^('2)` `a^2+b^2=a^('2)+b^('2)` (d) none of these

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

Explanation

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

1. \( ax + by + c = 0 \)
2. \( a'x + b'y + c = 0 \)
3. \( ax + by + c' = 0 \)
4. \( a'x + b'y + c' = 0 \)

### Step 1: Understanding the condition of perpendicular diagonals
For a quadrilateral to have perpendicular diagonals, it can be shown that the product of the slopes of the diagonals must equal -1. However, in this case, we can also use the property of the distances between parallel lines.

### Step 2: Finding the distance between parallel lines
The distance \( d \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

### Step 3: Calculate distances for the given lines
For the lines \( ax + by + c = 0 \) and \( ax + by + c' = 0 \):
- The distance \( d_1 \) is:

\[ d_1 = \frac{|c' - c|}{\sqrt{a^2 + b^2}} \]

For the lines \( a'x + b'y + c = 0 \) and \( a'x + b'y + c' = 0 \):
- The distance \( d_2 \) is:

\[ d_2 = \frac{|c' - c|}{\sqrt{a'^2 + b'^2}} \]

### Step 4: Setting the distances equal
Since the diagonals are perpendicular, the distances must be equal, so we set \( d_1 = d_2 \):

\[ \frac{|c' - c|}{\sqrt{a^2 + b^2}} = \frac{|c' - c|}{\sqrt{a'^2 + b'^2}} \]

Assuming \( |c' - c| \neq 0 \), we can cancel it from both sides:

\[ \frac{1}{\sqrt{a^2 + b^2}} = \frac{1}{\sqrt{a'^2 + b'^2}} \]

### Step 5: Squaring both sides
Squaring both sides gives:

\[ a^2 + b^2 = a'^2 + b'^2 \]

### Conclusion
Thus, we have shown that:

\[ a^2 + b^2 = a'^2 + b'^2 \]

This corresponds to option (c).

### Final Answer
The correct option is:
**(c) \( a^2 + b^2 = a'^2 + b'^2 \)**