If $x, y \in \mathbb{R}$ then the equation $3x^2 - 2(9y + 8)x^2 + (361y^2 + 2(100 + y^3)x + 64) = 2(190y + 2y^2)$ represents in rectangular Cartesian system:
Step-by-Step Solution
Key Concept: A sum of squares equals zero if and only if each squared term is individually zero, yielding multiple conic sections.
Expanding and simplifying $3x^4-2(19y+8)x^2+[(19y^2)+(10)^2+(10)^2+y^4+y^4+8^2]=2(19\times 10y+10y^2-8y^2)$ leads to $[x^2-(19y-10)]^2+[x^2-(10-y^2)]^2+[x^2-(y^2+8)]^2=0$. This gives three simultaneous equations: $x^2=19y-10$, $x^2=10-y^2$, and $x^2=y^2+8$, representing a parabola, circle, and hyperbola respectively.
Correct Answer: 1,2,3