Question

Identify the quadric surface. $$ \frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1 $$

Short answer

The given quadric surface equation represents an ellipsoid.

Step-by-step solution

Identify the form of the equation

Identify which type of standard quadric surface equation the given equation matches. Most quadric surfaces can be expressed in a standard form, where the squared variables are all added or subtracted and each divided by a constant squared. So, looking at the given equation \(\frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1\) , all variables have degree 2, all are added together and are each divided by a squared constant, which is the standard form for an ellipsoid.

Confirming the identification of the quadric surface

Now, confirm whether our identification matches the definition of an ellipsoid. By definition, an ellipsoid is a quadric surface for which all squared terms are added together and their coefficients are all positive. This matches exactly the provided equation.