Problem 41 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The given equation represents an ellipsoid.

Step 1: Identification

Evaluate the given equation, \(x^{2}+\frac{y^{2}}{4}+z^{2}=1\), and compare it with the standard equations of quadric surfaces such as ellipsoids, hyperboloids, parabolic cylinders and so on. We are looking for an equation where squares of all variables exist and are added together, and these squares are equal to some constant.

Step 2: Classification

Any expression that has the form \(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}=1\), where a, b, and c are some constants, represents an ellipsoid. Since the given equation \(x^{2}+\frac{y^{2}}{4}+z^{2}=1\) matches this form, it represents an ellipsoid.

Step 3: Explanation of the Result

An ellipsoid is a three-dimensional geometric figure whose plane sections are either ellipses or circles. The given equation matches the standard form and all terms are positive, so it represents an ellipsoid.

Key Concepts

Ellipsoid

An ellipsoid is a special type of quadric surface in three-dimensional geometry. It is characterized by an equation where the squares of variables are combined and are equal to a constant. Let's look at the general form of an ellipsoid equation:

• \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]
• Here, \(a\), \(b\), and \(c\) are the semi-axis lengths of the ellipsoid along the x, y, and z axes, respectively.

The shape of an ellipsoid can be compared to a "squished" or "stretched" sphere. If all three semi-axes are equal, it is a sphere.
For our problem, the equation is given as \(x^2 + \frac{y^2}{4} + z^2 = 1\). This matches the ellipsoid pattern with \(a = 1\), \(b = 2\) (since \(b^2 = 4\)), and \(c = 1\). Hence, this is indeed an ellipsoid, representing how these dimensions change its geometry.

Three-Dimensional Geometry

Three-dimensional geometry involves shapes that have depth in addition to height and width. Quadric surfaces like ellipsoids exist in this space and can be represented by equations involving three variables.
Within the world of three-dimensional shapes, an ellipsoid is among several types of quadric surfaces. These include:

• Ellipsoids (the focus of our current exercise)
• Hyperboloids
• Paraboloids
• Cylinders

Each of these forms has its own distinct equation and shape, providing a rich diversity of structures that can be analyzed in three dimensions. Recognizing an ellipsoid involves identifying equations with added squares of variables, signifying the shape's presence in a three-dimensional context.

Equation Identification

Identifying the type of a quadric surface given an equation is crucial for understanding its geometric form. In our exercise, we start with the equation \(x^2 + \frac{y^2}{4} + z^2 = 1\).
The key to identification is comparing this equation with known standard forms of quadric surfaces. Looking at the equation:

• The presence of squared terms \(x^2\), \(\frac{y^2}{4}\), and \(z^2\) indicates symmetry and balance.
• The absence of products of different variables (like \(xy\) or \(yz\)) means no cross-terms are present.
• All terms added to equal 1 suggest a surface closed like an ellipsoid.

By recognizing these patterns, we can accurately determine that the equation indeed represents an ellipsoid based on its squared-term structure. This process is critical for identifying and understanding geometric properties of surfaces in mathematical and real-world applications.