Question

In Exercises \(11-18,\) identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ z^{2}=x^{2}+\frac{y^{2}}{4} $$

Short answer

The quadric surface described by the equation \(z^{2}=x^{2}+\frac{y^{2}}{4}\) is an Elliptic Cone.

Step-by-step solution

Identify the Coefficients

The coefficients are \(1\) for \(x^2\), \(\frac{1}{4}\) for \(y^2\), and \(1\) for \(z^2\). These help to identify the type of the quadric surface.

Identify the Quadric Surface

Using the standard form for an Elliptic Cone, which matches the given equation, we can determine that the given equation represents an Elliptic Cone.

Sketch the Quadric Surface

Knowing that this is an elliptical cone, draw an inverted cone on one side of the \(z-\)axis and a regular cone on the other side of the \(z-\)axis to represent both 'nappe' of the cone. The wider spread of the cone should be along the \(y-\)axis due to the smaller coefficient of \(y^2\). The tips of the cones meet at the origin.

Confirm the Sketch Using a Computer Algebra System

Using a computer algebra system like Mathematica or Desmos, the quadric surface equation can be input to verify the sketch.