Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Identify the quadric surface. $$ z^{2}-x^{2}-\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The given equation represents a hyperboloid of two sheets where the z-axis is the axis of the hyperboloid.

Step 1: Identify the type of the equation

For the given equation \(z^{2}-x^{2}-\frac{y^{2}}{4}=1\), all variables are squared which indicates a possibility for the surface to be an ellipsoid, hyperboloid of one sheet, or hyperboloid of two sheets. To decide which one it is, we need to look at the coefficients and signs of the terms.

Step 2: Look at the signs

An ellipsoid has all positive squares, a hyperboloid of one sheet has one negative square and the rest positive, a hyperboloid of two sheets has two negative squares and one positive square. In our equation, we have two negative squares and one positive, which indicates that our quadric surface is a hyperboloid of two sheets.

Step 3: Attain the conclusion

The given equation is a hyperboloid of two sheets because we have two negative square terms and one positive square term. The variable with the positive square term will be the axis of the hyperboloid.