Problem 14 - University Calculus: Early Transcendentals

Question

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+z^{2}=4, \quad y=x$$

Step-by-Step Solution

Answer

The intersection is an ellipse in the plane where \(y = x\).

Step 1: Interpret the Equation of a Sphere

The equation \(x^2 + y^2 + z^2 = 4\) represents a sphere in 3-dimensional space. This sphere is centered at the origin \((0, 0, 0)\) with a radius of 2.

Step 2: Recognize the Plane Equation

The equation \(y = x\) represents a plane in the 3-dimensional space. This plane contains points where the values of \(x\) and \(y\) are equal, forming a diagonal in the \(xy\)-plane.

Step 3: Find the Intersection of the Sphere and the Plane

To find the geometric description, we need to determine where this plane intersects with the sphere. Substituting \(y = x\) into the sphere's equation gives \(x^2 + x^2 + z^2 = 4\), which simplifies to \(2x^2 + z^2 = 4\). This describes a new geometric shape within the plane.

Step 4: Simplify the Intersection Equation

Rearrange the equation from Step 3: \(2x^2 + z^2 = 4\). Divide the entire equation by 2 to get \(x^2 + \frac{z^2}{2} = 2\). This is the equation of an ellipse.

Step 5: Describe the Geometry of the Intersection

The intersection of the sphere and the plane \(y = x\) is an ellipse within the plane \(y = x\). The center of this ellipse remains at the origin, and it lies obliquely in the coordinate system, oriented along the direction of the plane \(y = x\).

Key Concepts

Sphere

A sphere is one of the simplest geometric shapes in three-dimensional space. It is defined by all points that are equidistant from a single point known as the center. The equation for a sphere centered at the origin \((0, 0, 0)\) is \(x^2 + y^2 + z^2 = r^2\), where \(r\) is the radius of the sphere. In our exercise, the equation \(x^2 + y^2 + z^2 = 4\) means we're dealing with a sphere that has * a center at \((0, 0, 0)\) * a radius of 2, since \( ext{radius} = \sqrt{4} = 2\).
Understanding the sphere is crucial because it helps us visualize a closed and continuous surface in space. Every point on this sphere lies exactly 2 units away from the center, forming a perfectly round structure in all directions.

Plane Equation

A plane in three-dimensional space can be defined by an equation involving its coordinates. In this context, the equation \(y = x\) describes a plane where the \(y\) coordinate equals the \(x\) coordinate. This implies that it forms a diagonal within the \(xy\)-plane.
To imagine this, think about drawing a 45-degree line in the flat, two-dimensional \(xy\)-plane on paper. Now extend that line infinitely in both directions and into the third dimension, \(z\), creating a plane that slices through space. This plane effectively divides space into regions where \(y\) values are either greater than or less than \(x\) values, and our specific plane is where they're exactly equal.
Understanding this plane's position helps in determining where it might intersect other shapes, such as the sphere mentioned.

Ellipse

An ellipse is an oval shape that can be seen as a stretched circle. It appears commonly in geometry when slicing through a cylinder or a sphere. In our scenario, the intersection of the sphere \(x^2 + y^2 + z^2 = 4\) and the plane \(y = x\) forms an ellipse.
By substituting \(y = x\) into the sphere's equation, the problem simplifies to the form \(2x^2 + z^2 = 4\). Dividing each part by 2 allows us to express this in the more recognizable ellipse equation: \(x^2 + \frac{z^2}{2} = 2\).
This ellipse is centered at the origin \((0, 0, 0)\), but lies within the plane \(y = x\). Unlike a circle, which has uniform radius, the ellipse has different lengths for its semi-major and semi-minor axes, providing that elongated appearance. Identifying this geometric shape provides deeper insight into understanding intersections in space.