Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Describe the traces of the surface in the given planes. $$ y^{2}+z^{2}-x^{2}=1 \quad x y \text { -plane, } x z \text { -plane, } y z \text { -plane } $$

Step-by-Step Solution

Answer

In xy-Plane, the surface traces a hyperbola with the equation \(y^{2} - x^{2} = 1\). Similarly, in xz-plane, it traces a rotated hyperbola with the equation \(-x^{2} + z^{2} = 1\). However, on the yz-plane, the surface traces a circle with the equation \(z^{2} + y^{2} = 1\).

Step 1: Traces in xy-plane

Set \(z = 0\) in the equation. This results in a hyperbola equation \(y^{2} - x^{2} = 1\).

Step 2: Traces in xz-plane

Set \(y = 0\) in the equation. This results in a hyperbola equation \(- x^{2} + z^{2} = 1\). This is a rotated version of the normal hyperbola \(x^{2} - y^{2} = 1\).

Step 3: Traces in yz-plane

Set \(x = 0\) in the equation. This results in a circle equation \(y^{2} + z^{2} = 1\).

Key Concepts

Traces

When studying surfaces in three dimensions, it's often helpful to analyze what the surface looks like in two dimensions on specific planes. This process is known as finding the 'traces' of a surface. By computing these traces, we gain insights into the behavior of the surface in different sections.
For example:

• xy-plane: By setting \( z = 0 \), traces are found in the plane where the coordinate is zero. In this exercise, by substituting \( z = 0 \) into the given equation, we simplify to \( y^2 - x^2 = 1 \).
• yz-plane: By setting \( x = 0 \), this trace simplifies the examination of the surface on this plane. Substituting \( x = 0 \) results in \( y^2 + z^2 = 1 \), revealing a circle.
• xz-plane: Here, by setting \( y = 0 \) in the equation, you obtain another trace equation \( -x^2 + z^2 = 1 \), illustrating another orientation of a hyperbola.

Traces are crucial for visualizing and understanding the spatial structure of complex surfaces, simplifying an otherwise daunting task.

Hyperbola

A hyperbola is a type of conic section that occurs when a plane intersects both halves of a double cone. This interaction forms two symmetrical open curves. The standard equation for a hyperbola in its simplest form looks like:\[ x^2 - y^2 = 1 \]In our exercise, the equation \( y^2 - x^2 = 1 \) derived from the xy-plane and \( -x^2 + z^2 = 1 \) from the xz-plane both describe hyperbolas that are oriented based on the traces:

• Open curves: Hyperbolas consist of two unbounded branches that open indefinitely, defining their unique shape.
• Center and axes: The point midway between the curves of a hyperbola is the center. Axes determine their orientation and distances from the center to each curve.
• Orientation: Hyperbolas can be rotated, as demonstrated by the two equations we've encountered.

Understanding the hyperbola's unique properties and how they present in different planes broadens conceptual comprehension of three-dimensional surfaces.

Circle

The circle is another fundamental conic section, characterized by a curve where all points are equidistant from a specific point called the center. In the yz-plane trace, setting \( x = 0 \) in our equation yields a classic circle described by \( y^2 + z^2 = 1 \). Key features of a circle include:

• Center: In our trace, the center is at the origin (0,0) in the yz-plane.
• Radius: In this scenario, the radius is 1 since all points obey the equation \( y^2 + z^2 = 1 \).
• Uniformity: Unlike hyperbolas or ellipses, circles maintain constant curvature and equidistance from the center, providing a symmetrical form.

The circle's structure simplifies visualization and allows comprehension of the spherical qualities present in spatial graphs or cross-sections.