Problem 51 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The given quadric surface is an elliptic cone.

Step 1: Recognize The Form

The general form of the equation of an elliptic cone in standard position is given by \(z^{2}=Ax^{2}+By^{2}\) where A and B are constants. So, our equation can be written as \(z^{2}=Ax^{2}+By^{2}\) where A = 9 and B = 1.

Step 2: Identify The Surface

The given equation, \(z^{2}=9x^{2}+y^{2}\), can be rewritten as \((z/3)^{2}=(x^{2})+(y/3)^{2}\) which is the equation for an elliptic cone in standard form. Since we have a z^2 term and the sum of the squares of x and y, this is an elliptic cone.

Key Concepts

Quadric Surfaces

Quadric surfaces are a fascinating topic in mathematics. They are the 3D extensions of conic sections and include various shapes. Quadric surfaces are defined by second-degree polynomial equations of the form:

• Axes aligned: Each variable is raised to a power of two and combined linearly.
• Variety: Includes ellipsoids, hyperboloids, paraboloids, and more.
• Symmetry: Often symmetric about one or more coordinate planes.

For example, an equation like \(Ax^2 + By^2 + Cz^2 = 1\) represents an ellipsoid. Depending on how the variables and constants are arranged, you can also get hyperboloids or paraboloids. To identify or graph these surfaces, pay close attention to the equation structure and coefficients. They determine the type and orientation of the surface.

Standard Form Equation

Understanding the standard form of equations is vital in identifying quadric surfaces. The standard form of a quadric surface equation eliminates terms and arranges the remaining variables:

• Format: Arranged to easily identify the constant coefficients.
• Reduction: Uses algebraic manipulation to identify simpler forms.
• Clarity: Makes recognizing the type of surface easier.

In our context, the elliptic cone equation \(z^{2}=Ax^{2}+By^{2}\) is already in a standard form. This form helps distinguish elliptic cones from other surfaces because it emphasizes the relationship between the variables with no mixed terms. By aligning the equation to its standard form, we can quickly see if and how terms relate or differ, making it easier to visualize or compute the surface.

Elliptic Cone Equation

Elliptic cone equations are a specific type of quadric surface. They are unique because they create a cone-like shape but with elliptical cross-sections rather than circular. Here's what characterizes an elliptic cone:

• Structure: Given by the equation \(z^2 = Ax^2 + By^2\).
• Coefficients: The constants \(A\) and \(B\) dictate the stretch and type of ellipsis.
• Symmetry: Perfectly symmetric about the z-axis.

An elliptic cone opens up or down along the z-axis, showing its symmetry. The relationship among \(A\), \(B\), and \(z^2\) means it expands outward from the z-intercept, offering a clear method to recognize it among quadric surfaces.In our original exercise, the equation \(z^{2}=9x^{2}+y^{2}\) is straightforward. By identifying \(A=9\) and \(B=1\), it's evident that it's an elliptic cone. Transforming the variables to see this as a standard form highlights its elliptical nature in cross-sections through the cone.