Prove that the equation m `(x^3-3xy^2)+y^3-3x^2y=0` represents three straight lines equally inclined to each other.

Options

A. they are equally inclined to one another
B. two of which are at right angles
C. two of which are coincident
D. none of these

Answer

A

Explanation

Let `y=mx` be one of the lines represented by the given equation. Then, `y=mx` satisfies it.
`therefore" "lambda(1-3m^(2))+m^(3)-3m=0`
`rArr" "lambda=(3m-m^(3))/(1-3m^(2))`
`rArr" "tan phi=(3tantheta- tan^(3)theta)/(13tan^(2)theta)`, where `lambda= tan phi and m = tan theta`
`rArr" "tan phi=tan 3 theta`
`rArr" "3 theta=phi, pi + phi, 2pi + phi`
`rArr" "theta=(phi)/(3),(pi)/(3)+(phi)/(3),(2pi)/(3)+(phi)/(3)`
Hence, the given equation represents three lines passing through the origin such that they are equally inclined with one another.