Final answer:
The equation x² + 6y - 6z² = 0 can be rearranged to y = -(x²)/6 + z², which represents a hyperbolic paraboloid, a saddle-shaped surface commonly used in architecture.
Step-by-step explanation:
The equation provided by the student, x² + 6y - 6z² = 0, resembles a standard quadric surface equation, which can be transformed into one of the standard forms of conic sections. To classify this surface, we need to rearrange and compare it with the standard forms of quadratic surfaces.
First, we can rewrite the equation as follows:
x² - 6z² + 6y = 0
Rearranging terms gives:
x² - 6z² = -6y or y = -(x²)/6 + z²
Since the coefficients of x² and z² have opposite signs and the equation is solved for y, this surface is a hyperbolic paraboloid. A hyperbolic paraboloid has a saddle shape with one axis as a hyperbola and another as a parabola. This type of surface is often used in architecture and design due to its unique structural properties.
Unfortunately, we cannot sketch the surface here. However, a rough sketch of a hyperbolic paraboloid would show a curve opening upwards along the y-axis, curving downward along the x-axis, and bending in the opposite direction along the z-axis, creating a saddle-shaped surface.