91Ó°ÊÓ

We can rewrite the given equation as: $$\frac{x^2}{6} + \frac{z^2}{16} = y$$ This equation is similar to the general equation of an elliptic paraboloid: $$\frac{x^2}{a^2} + \frac{z^2}{c^2} = y$$ So, the surface defined by the given equation is an elliptic paraboloid.

The main features of an elliptic paraboloid include the vertex, the axis of symmetry, and the direction of the opening. 1. Vertex: The vertex of the elliptic paraboloid is the point where the surface touches or crosses the plane y = 0. In our case, this occurs when x = 0 and z = 0. Therefore, the vertex is at the origin, V(0, 0, 0). 2. Axis of symmetry: The axis of symmetry is the line passing through the vertex and perpendicular to the plane of the paraboloid's base ellipse. In this case, the axis is the y-axis, along which it exhibits rotational symmetry. 3. Direction of the opening: Since the constants a^2 and c^2 in the equation are both positive and y is on one side of the equation, the paraboloid opens upward (along the positive y-axis).

The surface defined by the equation $$y=\frac{x^2}{6}+\frac{z^2}{16}$$ is an elliptic paraboloid. Its vertex is at the origin (0, 0, 0), it is symmetric around the y-axis, and it opens upward along the positive y-axis.