Final answer:
The equation of the conical surface with its vertex at the origin and a lemniscate as the guiding curve is (x²+y²)²=(z²/c²)(x²-y²), resulting from scaling the lemniscate based on the z-coordinate.
Step-by-step explanation:
The student question pertains to finding the equation of a conical surface whose vertex is at the origin and which uses the lemniscate given by the equation (x²+y²)²=c²(x²-y²) as its guiding curve at the level z=c. To determine the equation of the conical surface, we exploit the similarity of cross-sections at different heights. Since the guiding curve is at z=c, the equation of the conical surface corresponding to each cross-section will be a scaled version of the guiding curve, adjusted by the ratio z/c. Thus, the equation of the conical surface is (x²+y²)²=(z²/c²)(x²-y²), which defines the locus of points (x, y, z) that form the conical surface.