If diagonals of the quadrilateral formed by lines $ax+by+c=0$; $a'x+b'y+c=0$; $ax+by+c'=0$ and $a'x+b'y+c'=0$ are perpendicular, then which of the following is correct?

Question

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the conditions under which the diagonals of the quadrilateral formed by the given lines are perpendicular.

The lines given 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 $

These lines form a quadrilateral, and the diagonals of this quadrilateral are formed by the intersection of opposite sides. The diagonals are:
- The line joining the intersection of lines 1 and 4 with the intersection of lines 2 and 3.
- The line joining the intersection of lines 1 and 3 with the intersection of lines 2 and 4.

For the diagonals to be perpendicular, the dot product of their direction vectors must be zero.

### Explanation for Option 2:

The direction vectors of the diagonals can be derived from the coefficients of the lines. For the diagonals to be perpendicular, the condition is that the sum of the squares of the coefficients of one diagonal must equal the sum of the squares of the coefficients of the other diagonal. This condition is mathematically expressed as:

$$ a^2 + b^2 = a'^2 + b'^2 $$

This is precisely what Option 2 states. Therefore, Option 2 is correct because it reflects the condition for the diagonals to be perpendicular based on the coefficients of the lines forming the quadrilateral.

### Explanation for why other options are incorrect:

1. **Option 1: $ b^2 + c^2 = {b'}^2 + {c'}^2 $**
   - This option suggests a relationship involving the coefficients $ b $ and $ c $, and their counterparts $ b' $ and $ c' $. However, the perpendicularity of the diagonals is determined by the coefficients of the lines in terms of $ a $ and $ b $, not $ c $. Therefore, this option does not correctly represent the condition for perpendicular diagonals.

2. **Option 3: $ c^2 + a^2 = {c'}^2 + {a'}^2 $**
   - Similar to Option 1, this option involves $ c $ and $ a $. The perpendicularity condition is not related to the constant terms $ c $ and $ c' $ in the equations of the lines. Thus, this option does not correctly describe the condition for perpendicular diagonals.

3. **Option 4: $ b^2 - c^2 = {b'}^2 - {c'}^2 $**
   - This option suggests a difference of squares involving $ b $ and $ c $. However, the condition for perpendicular diagonals is based on the equality of the sums of squares of the coefficients $ a $ and $ b $, not differences. Therefore, this option is incorrect.

In conclusion, Option 2 is the correct answer because it accurately represents the condition for the diagonals of the quadrilateral to be perpendicular, based on the equality of the sums of squares of the coefficients $ a $ and $ b $.