Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Identify the quadric surface. $$ z^{2}=2 x^{2}+2 y^{2} $$

Step-by-Step Solution

Answer

The given quadric surface \(z^{2}=2 x^{2}+2 y^{2}\) represents a cone.

Step 1: Identify the Quadric Surface Form

List down the standard forms of quadric surfaces and compare them with the given equation. Quadric surfaces include ellipsoids, hyperboloids, paraboloids, cones, and planes. In our case, \(z^{2}=2 x^{2}+2 y^{2}\) is similar to the standard form of a cone given by \(z^{2} = a^2x^2 + b^2y^2\) where \(a=b= \sqrt{2}\)

Step 2: Confirm the Result

To confirm that the given equation is a cone, take a cross-section along the z-axis. This would slice the 3D surface at a given z, yielding a 2D curve in the xy-plane. For the equation \(z^{2}=2 x^{2}+2 y^{2}\), if we make z constant, we get a circle in the xy-plane, which corresponds to a cone's cross-section. This confirms that the surface is indeed a cone.