The lines represented by the equation $Ax^2 + 2Bxy + Hy^2$ = 0 are perpendicular, if :

Question

Accepted Answer

To determine when the lines represented by the equation $ Ax^2 + 2Bxy + Hy^2 = 0 $ are perpendicular, we need to analyze the condition for perpendicularity in the context of conic sections, specifically degenerate conics that represent pairs of lines.

### Explanation for Option 4: $ A + H = 0 $

The given equation $ Ax^2 + 2Bxy + Hy^2 = 0 $ represents a pair of lines. For these lines to be perpendicular, the condition $ A + H = 0 $ must be satisfied. This is derived from the general condition for perpendicularity of lines in the form of a conic section:

1. **Perpendicular Lines Condition**: For the lines represented by the equation $ Ax^2 + 2Bxy + Hy^2 = 0 $ to be perpendicular, the sum of the coefficients of $ x^2 $ and $ y^2 $ must be zero. This is because the angle between the lines is determined by the coefficients of the quadratic terms. Specifically, the condition $ A + H = 0 $ ensures that the angle between the lines is 90 degrees, making them perpendicular.

### Explanation for Why Other Options are Incorrect

1. **Option 1: $ AH = -1 $**
   - This condition does not relate to the perpendicularity of the lines. Instead, it might relate to other properties of the conic, such as the nature of the conic section (ellipse, hyperbola, etc.), but not specifically to the perpendicularity of the lines.

2. **Option 2: $ A + B = 0 $**
   - This condition does not ensure that the lines are perpendicular. It involves the coefficient $ B $, which is associated with the $ xy $ term and affects the orientation of the lines but not directly their perpendicularity.

3. **Option 3: $ B + H = 0 $**
   - Similar to Option 2, this condition involves the coefficient $ B $ and does not directly relate to the perpendicularity of the lines. It might affect the symmetry or orientation of the lines but not their perpendicularity.

In conclusion, the correct condition for the lines represented by the equation $ Ax^2 + 2Bxy + Hy^2 = 0 $ to be perpendicular is $ A + H = 0 $, as stated in Option 4. This condition directly ensures that the angle between the lines is 90 degrees, fulfilling the requirement for perpendicularity.