Question

Describe the traces of the surface in the given planes. $$ \frac{x^{2}}{4}+y^{2}+z^{2}=1 \quad x y \text { -plane, } x z \text { -plane, } y z \text { -plane } $$

Short answer

The traces of the given ellipsoid in the xy-plane, xz-plane, and the yz-plane are the ellipses \( \frac{x^{2}}{4}+y^{2}=1 \) and \( \frac{x^{2}}{4}+z^{2}=1 \) as well as the circle \(y^{2}+z^{2}=1 \).

Step-by-step solution

Trace in xy-plane

Set z=0 in the given equation, this will result in the equation for the trace in the xy-plane. This gives us the equation \( \frac{x^{2}}{4}+y^{2}=1 \), which is an ellipse with semi-major axis \(a=1\) and semi-minor axis \(b=2\).

Trace in xz-plane

Set y=0 in the given equation, this will result in the equation for the trace in the xz-plane. This gives us the equation \( \frac{x^{2}}{4}+z^{2}=1 \), which is an ellipse with semi-major axis \(a=1\) and semi-minor axis \(b=2\).

Trace in yz-plane

Set x=0 in the given equation, this will result in the equation for the trace in the yz-plane. This gives us the equation \(y^{2}+z^{2}=1 \), which is a circle with radius 1.

Key concepts

Ellipses in Coordinate Planes

When we encounter equations that represent shapes in coordinate systems, understanding how to visualize and analyze these shapes is key. An ellipse is a common shape that can appear when slicing through a three-dimensional object in a coordinate plane. To define an ellipse within a coordinate plane, we often come across equations of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively.

For example, if we set one of the variables to zero, like assigning \(z=0\) to focus on the \(xy\)-plane, the equation simplifies by removing the variable that corresponds to the axis we're ignoring. In the exercise, when \(z=0\), the equation becomes \( \frac{x^2}{4} + y^2 = 1 \) indicating an ellipse in the \(xy\)-plane with a semi-major axis of 2 and a semi-minor axis of 1. To visualize this, imagine stretching a circle along the \(x\)-axis to form the ellipse.

When analyzing textbook solutions, rather than just noting the steps, it's beneficial to grasp why setting \(z=0\) yields the two-dimensional representation of the ellipse and how the constants in the equation relate to its measurements. This understanding can help students envision ellipses in other exercises and apply the concept to different coordinate planes.

Circle in Coordinate Plane

A circle is a special type of ellipse with equal radius along all axes. To find a circle within a three-dimensional object when intersecting with a coordinate plane, we often see equations like \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. The beauty of a circle lies in its symmetry, which provides simplicity when representing it in any plane.

In our exercise, setting \(x=0\) reduces the given equation to \(y^2 + z^2 = 1\), identifying a locus of points in the \(yz\)-plane that forms a circle with radius 1. This step in the solution demystifies how a complex 3D surface can intersect with a plane to reveal a perfect circle. Envisaging a circle on the \(yz\)-plane or any other plane entails fixing one axis and letting the remaining variables define the circular boundary in that plane.

Enhancing textbook solutions by describing the process of simplification and visualization aids in solidifying the student's understanding. Recognizing that the equation of a circle in any coordinate plane maintains the same form, with differing variables, equips learners with the knowledge to tackle similar problems with confidence.

3D Surface Traces

Three-dimensional surfaces can be complex to visualize, but understanding their traces—cross sections formed when slicing the surface by a plane—can immensely aid our comprehension. Traces simplify 3D surfaces into familiar 2D shapes, depending on the orientation of the slicing plane. In the context of our exercise, we examine a 3D ellipsoid represented by the equation \(\frac{x^2}{4} + y^2 + z^2 = 1\).

By setting one of the variables to zero, such as \(x=0\), \(y=0\), or \(z=0\), we 'slice' the ellipsoid to produce the traces in the respective planes. These traces reveal the shapes of ellipses or circles that lie in the \(xy\)-, \(xz\)-, or \(yz\)-planes, respectively. The key takeaway for students is that these cross-sectional views provide a strategic approach to dissecting a complicated object into simpler, more analyzable pieces.

To enhance textbook solutions, embedding this conceptual understanding within the step-by-step process gives students a more holistic grasp of the subject matter. It's not just about finding the intersections but also about building a three-dimensional awareness of how 2D shapes fit within a larger 3D puzzle. Breaking down a formidable topic like 3D surface traces into comprehensible segments makes the learning process both manageable and enlightening.