Question

The two of the straight lines represented by the equation ax³ + bx²y + cxy² + dy³ = 0 will be right angle if-

• A. a² + c² = 0
• B. a² + ac + bd + d² = 0
• C. a²c² + bd + d² = 0
• D. None of these

Answer 1

Answer: (B)

a² + ac + bd + d² = 0

Answer 2

Let y = mx be any line represented by the equation

ax² + bx² y + cxy² + dy³ = 0

Ž ax³ + bx² (mx) + cx (m²x²) + dm³x³ = 0

Ž a + bm + cm² + dm³ = 0 which is a cubic equation

It represents three lines out of which two are perpendicular hence :

m₁ m₂ = – 1 and m₁m₂m₃ = $\frac { - \mathrm { a } } { \mathrm { d } }$ Ž m₃ =

and m₃ is the root of the given equation

hence a + b $\left( \frac { \mathrm { a } } { \mathrm { d } } \right)$ + c $\left( \frac { \mathrm { a } } { \mathrm { d } } \right) ^ { 2 }$ + d = 0

d² + bd + ca + a² = 0