Question

Identify the quadric surface. $$ 9 x^{2}+4 y^{2}-8 z^{2}=72 $$

Short answer

The given equation represents a hyperboloid of one sheet.

Step-by-step solution

Normalize the Equation

The first step in identifying the quadric surface of an equation is to normalize it by dividing all terms by the constant on the right side of the equation. For \( 9x^{2}+4y^{2}-8z^{2}=72 \), divide each term by 72 to get \( \frac{x^{2}}{8}+\frac{y^{2}}{18}-\frac{z^{2}}{9}=1 \)

Identify the Quadric Surface

With the normalized form of the equation, it can now be matched to one of the standard forms of quadric surfaces. Given the negative sign on one of the squared variables, it suggests that the equation represents a hyperboloid. In this case, because there are positive terms in the \(x^{2}\) and \(y^{2}\) positions, and a negative term in the \(z^{2}\) position, the equation specifically represents a hyperboloid of one sheet.