Problem 39 - University Calculus: Early Transcendentals

Question

Describe the given set with a single equation or with a pair of equations. The circle of radius 2 centered at (0,2,0) and lying in the a. \(x y\) -plane b. \(y z\) -plane c. plane \(y=2\)

Step-by-Step Solution

Answer

a. \(x^2 + (y-2)^2 = 4\) b. \((y-2)^2 + z^2 = 4\) c. \(x^2 + z^2 = 4\)

Step 1: Identify circle properties

The circle has a radius of 2 and is centered at the point (0,2,0). We know that the general equation of a circle with center \((h, k, l)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\).

Step 2: Circle equation for the xy-plane

For a circle in the \(xy\)-plane, the \(z\)-coordinate is constant. Since the center of the circle is \( (0, 2, 0) \), and it lies in the \(xy\)-plane, the circle equation is \((x - 0)^2 + (y - 2)^2 = 2^2\) so, \(x^2 + (y - 2)^2 = 4\).

Step 3: Circle equation for the yz-plane

In the \(yz\)-plane, the \(x\)-coordinate is constant. Thus, with the center at \( (0, 2, 0) \), the circle equation becomes \((y - 2)^2 + z^2 = 2^2\). So, the equation is \((y - 2)^2 + z^2 = 4\).

Step 4: Circle equation for the y=2 plane

Within the plane \(y = 2\), the \(y\)-coordinate is constantly 2. The center of the circle is still \( (0, 2, 0) \), and with the radius 2, the circle equation is \(x^2 + z^2 = 2^2\). Therefore, the equation is \(x^2 + z^2 = 4\).

Key Concepts

Circle Equations

When working with circles in 3D geometry, it is important to understand how to express them mathematically. The general formula for a circle in a three-dimensional space is \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]where \( (h, k, l) \) is the center of the circle and \[r\] is its radius. This equation is a variation of the circle equation in two dimensions.

• For a circle centered at a point in space, adjust the equation to match the specific location and size of the circle.
• This equation allows you to easily manipulate and find the circle's geometry in different coordinate planes.

Identifying which plane the circle lies in helps simplify this equation. For example, if a circle lies entirely within the \(xy\)-plane, the \(z\)-coordinate can be neglected, saving complexity in calculations.

Coordinate Planes

Coordinate planes are fundamental components in 3D geometry, and they refer to the two-dimensional slices of three-dimensional space.In three-dimensional space, there are three main planes:

• \(xy\)-plane: The plane where \(z=0\), often used to map two-dimensional coordinates in most 3D plots and known for being the base plane in computer graphics.
• \(yz\)-plane: This plane is situated where \(x=0\), often used in cross-sectional views in engineering and design.
• Plane \(y=2\): This is a less common plane that is parallel to the \(xz\)-plane but translated to \(y=2\).

Understanding which coordinate plane you are working in is crucial, as it determines which variables in your equations need to be constant or can be ignored altogether. This simplifies finding the circle's equation.

Radius and Center of Circle

The radius and center of a circle play crucial roles in defining its equation and understanding its position in space. The center is defined as a point \( (h, k, l) \), indicating the exact middle of the circle, which serves as a fixed reference point for its location.

• The radius, a scalar value \(r\), indicates the distance from the center to any point on the circle's circumference.
• Given a circle's radius and center, you can construct its equation whether it lies in the \(xy\)-plane, \(yz\)-plane, or any other specified plane.

For example, for a circle centered at \((0, 2, 0)\) with a radius of 2, in the \(xy\)-plane, the equation simplifies to \(x^2 + (y - 2)^2 = 4\). Knowing these parameters allows for straightforward visualization and manipulation of geometric shapes in three-dimensional spaces.