Question

The equation $x^3+x^{2}y-x{y}^2=y^3$ represents

• A. three real straight lines
• B. lines in which two of them are perpendicular to each other
• C. lines in which two of them are coincident
• D. None of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A three real straight lines
B lines in which two of them are perpendicular to each other
C lines in which two of them are coincident

$x^3-y^3+x^{2}y-x{y}^2=0$
$⟹(x-y)(x^2+y^2+xy)+xy(x-y)=0$
$⟹(x-y)(x^2+y^2+2xy)=0$
$⟹(x-y)(x+y{)}^2=0$
so roots are $x-y=0,x+y=0,x+y=0$
$x-y=0$ and $x+y=0$ are perpendicular to each other and two of them are coincident.