The equation `lambda(x^(3)-3xy^(2))+y^(3)-3x^(2)y=0` represent three straight lines passing through the origin such that

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 the above

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 tanphi=(3 tan theta-tantheta)/(1-3tan^2theta)" " [put m=tan theta and lambda = tan phi]`.
`rArr tan phi =tan 3 theta` . ltbr. `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 to one another.