The equation $x ^ { 3 } + y ^ { 3 } = x y ( 1 + x y )$ represents (a) Two parabolas intersecting at two points (b) Two parabolas touching at one point (c) Two non-intersecting hyperbolas (d) One parabola passing through the origin.
(A) $x ^ { 3 } + y ^ { 3 } = x y ( 1 + x y ) \Rightarrow ( y ^ { 2 } - x )( x ^ { 2 } - y ) = 0$. This is the equation of two parabolas $y ^ { 2 } = x$ and $x ^ { 2 } = y$, intersecting at $(0,0)$ and $(1,1)$.
The equation $x ^ { 3 } + y ^ { 3 } = x y ( 1 + x y )$ represents\\
(a) Two parabolas intersecting at two points\\
(b) Two parabolas touching at one point\\
(c) Two non-intersecting hyperbolas\\
(d) One parabola passing through the origin.