Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Identify the quadric surface. $$ 3 z=-y^{2}+x^{2} $$

Step-by-Step Solution

Answer

The given equation represents a hyperbolic paraboloid.

Step 1: Examine the Equation

To identify the type of quadric surface, you first need to examine the equation \( 3z = -y^2 + x^2 \). The equation can also be rearranged to bring it to a more standard form, as \( z = (-y^2 + x^2)/3 \).

Step 2: Identify the Quadratic Terms

Focus on the quadratic terms of the equation. It is clear that the signs are opposite, which will help identify the quadric surface type. When the signs of the square terms in the equation are opposite, it typically represents a hyperbolic paraboloid.

Step 3: Confirm Hyperbolic Paraboloid

The equation \( z = \frac{1}{3}(x^2 - y^2) \) confirms it is a hyperbolic paraboloid. In the standard form of the equation for a hyperbolic paraboloid \( z = \frac{1}{a^2}x^2 - \frac{1}{b^2}y^2 \), the coefficients of \( x^2 \) and \( y^2 \) are of opposite signs, just as in our given equation.