Question

The graph of equation $x+y=x^3+y^3$ is the union of-

• A. Line and an ellipse
• B. Line and a parabola
• C. Line and hyperbola
• D. Line and a point

Answer 1

Answer: (A)

Line and an ellipse

Given: $x+y=x^3+y^3$

$\Rightarrow x+y=(x+y{)}^3-3x^{2}y-3x{y}^2$

$\Rightarrow x+y=(x+y{)}^3-3xy(x+y)$

$\Rightarrow x+y=(x+y)\left[\right(x+y{)}^2-3xy]$

$\Rightarrow 0=(x+y)\left[\right(x+y{)}^2-3xy-1]$

$\Rightarrow 0=(x+y)\left[\right(x^2+y^2-xy-1]$

now, $I:x+y=0II:(x^2+y^2-xy-1)=0$

so, $I$ is a line and $II$ is an ellipse with axes rotated with some angle.