Problem 2 - Thomas Calculus

Question

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ z^{2}+4 y^{2}-4 x^{2}=4 $$

Step-by-Step Solution

Answer

It's a hyperboloid of one sheet.

Step 1: Identify the surface equation type

Observe the given equation: \( z^{2} + 4y^{2} - 4x^{2} = 4 \). Notice the presence of both positive and negative quadratic terms, which is indicative of a hyperboloid or hyperbolic paraboloid surface.

Step 2: Simplify the equation

Divide every term by the constant on the right side of the equation to rewrite it in a more familiar form:\[ \frac{z^2}{4} + y^2 - x^2 = 1 \] This equation suggests a hyperboloid.

Step 3: Classify the type of hyperboloid

Recognize that since there are two positive square terms and one negative, it signifies a hyperboloid of one sheet. A hyperboloid of one sheet typically has the form \( \frac{z^2}{a^2} + \frac{y^2}{b^2} - \frac{x^2}{c^2} = 1 \). Comparing this with our equation confirms it is a hyperboloid of one sheet.

Key Concepts

Surface Equation

The surface equation given in the exercise is \( z^{2} + 4y^{2} - 4x^{2} = 4 \). This type of equation is key in identifying the shape of a surface in three-dimensional space. Typically, surface equations in this context involve terms of the second degree, meaning variables are squared.By observing coefficients and signs in the equation, we can determine the type of surface it represents. In this case, the equation has both positive and negative terms, which is distinctive of surfaces such as hyperboloids or paraboloids.To decipher the specific type of surface, the equation can be manipulated to reflect its standard form. This involves isolating terms and potentially dividing the equation by a constant to simplify it, as shown in the solution steps. Dividing through by 4 transforms our equation into: \[ \frac{z^2}{4} + y^2 - x^2 = 1 \] Such transformations make it easier to see which specific type of surface the equation matches to.

Quadratic Surface

Quadratic surfaces are three-dimensional shapes defined by second-degree polynomial equations. They are a fundamental concept in both algebra and geometry. Some common types of quadratic surfaces include ellipsoids, paraboloids, and hyperboloids. For instance, the equation in our exercise, after simplification, represents a hyperboloid of one sheet. This is specifically identified by the combination of two positive quadratic terms and one negative term in its equation: - The presence of a negative term indicates that the surface extends in different directions. - The positive terms signify the symmetry of the shape along those axes. Recognizing these patterns is crucial for classifying the surface correctly. A quadratic surface like the hyperboloid is characterized by its curved yet infinite shape, making it extend outward indefinitely in form.

Three-Dimensional Analytic Geometry

Three-dimensional analytic geometry deals with the representation and properties of geometric objects in three-dimensional space using algebraic equations. It combines coordinates, algebra, and geometry to analyze spatial objects and phenomena.In this context, surfaces such as the hyperboloid can be represented and studied through precise mathematical equations like \( \frac{z^2}{4} + y^2 - x^2 = 1 \). Such equations allow us to understand how the surface behaves in space:- **Coordinate axes** help in visualizing and locating the surface in 3D space.- **Transformation techniques** simplify complex equations, making it easier to identify characteristics of the surface.- **Graphical interpretation** is possible, showing how these surfaces might look like, such as the gracefully extending sheets of a hyperboloid.By leveraging the principles of three-dimensional analytic geometry, we gain deeper insights into the form and structure of surfaces, making complex objects easier to understand and analyze.