Question

If ad ≠ 0 and two of the lines represented by \( ax^3 + 3bx^2y + 3cxy^2 + dy^3 = 0 \) are perpendicular, then:

• A. \( a^2 + ac + bd + d^2 = 0 \)
• B. \( a^2 + 3ac + 3bd + d^2 = 0 \)
• C. \( a^2 - 3ac - 3bd + d^2 = 0 \)
• D. \( a^2 + 3ac - 3bd + d^2 = 0 \)

Hint:

For perpendicular lines in geometry, use the condition that the product of their slopes is -1, or in higher-order equations, use the derived relations.

Answer 1

The Correct Option is B

For the lines to be perpendicular, the condition \( a^2 + 3ac + 3bd + d^2 = 0 \) must hold. This comes from the orthogonality condition of the lines represented by the homogeneous cubic equation.