Problem 29 - Thomas Calculus

Question

In Exercises \(25-34\) , 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 $$ \text { a. } x \text { y-plane } \quad \text { b. } y z \text { -plane } \quad \text { c. plane } y=2 $$

Step-by-Step Solution

Answer

a. \(x^2 + (y-2)^2 = 4, z = 0\); b. \((y-2)^2 + z^2 = 4, x = 0\); c. \(x^2 + z^2 = 4, y = 2\).

Step 1: Identify the Center and Radius

The given problem states that the circle has a radius of 2 and is centered at the point \((0, 2, 0)\). This information will be relevant in forming the equations for different planes.

Step 2: Equation for Circle in the xy-plane

In the xy-plane, the z-coordinate is constant. Given that the center is at \((0, 2, 0)\) and lies in the xy-plane, the constant z value is \( z = 0 \). Therefore, we use the equation for a circle in the xy-plane: \[(x - 0)^2 + (y - 2)^2 = 2^2\] This simplifies to: \[x^2 + (y - 2)^2 = 4\] Additionally, the equation \( z = 0 \) is needed because the circle is in the xy-plane.

Step 3: Equation for Circle in the yz-plane

In the yz-plane, the x-coordinate is constant. The circle has its center at \((0, 2, 0)\). Thus, the x-coordinate is \( x = 0 \). The circle's equation in the yz-plane is: \[(y - 2)^2 + z^2 = 4\] Additionally, the equation \( x = 0 \) is required because the circle lies in the yz-plane.

Step 4: Equation for Circle in the Plane y=2

For a circle in the plane where \( y = 2 \), the y-coordinate is constant. Hence, we analyze along the x and z axes. With the given center of \((0, 2, 0)\) and radius 2, the equation is: \[x^2 + z^2 = 4\]This represents a circle in the plane \( y = 2 \).

Key Concepts

Equations of circles

A circle is a set of all points that are at a specific distance, known as the radius, from a given point called the center. In geometry and calculus, understanding the equation of a circle is essential to accurately describe its location and size in a coordinate system. In its standard form, the equation of a circle centered at \((h, k)\) with a radius \(r\) is expressed as:\[(x - h)^2 + (y - k)^2 = r^2\]- **Center**: The point \((h, k)\) represents the center of the circle.- **Radius**: The radius \(r\) informs how large or small the circle is.For example, if we have a circle with a center at \((0, 2)\) and a radius of 2, its equation will be:\[x^2 + (y - 2)^2 = 4\]When dealing with multiple dimensions, such as 3D geometry, circles can also lie in different planes. The planes like the **xy-plane**, **yz-plane**, or another specific plane like **y = some constant** will influence the form of the circle's equation by fixing one coordinate, simplifying its expression.

Coordinate planes

Coordinate planes are foundational concepts in geometry and calculus, as they allow representations of geometric figures using algebraic equations. Each plane in a 3D space can be seen as a flat, two-dimensional surface extending infinitely: - **xy-plane**: Here, any point has a z-coordinate of 0. At this plane, the focus is on the x and y coordinates, making it easier to visualize and apply the circle's equation. - **yz-plane**: In this plane, the x-coordinate is fixed at 0. This enables the understanding and solving of equations using just y and z coordinates. - **Custom planes**, like **y = 2**: In such scenarios, the y-coordinate is constant, and the relationships between the x and z coordinates form the main consideration. Each of these planes effectively becomes a two-dimensional canvas where circles can be positioned and analyzed. By fixing one coordinate, such as x, y, or z, the calculations are limited to the remaining two dimensions, which simplifies solving equations or visualizing shapes.

3D geometry

3D geometry expands our understanding of shapes beyond the flat two-dimensional world. In three-dimensional space, geometric figures such as circles can exist on different planes and assume various orientations:- In **3D coordinate systems**, points are defined by three coordinates: (x, y, z). This allows for a richer representation of objects, such as circles, which can now move and interact within planes that are not just limited to the flat surfaces of traditional graph paper.- **Planes in 3D** could be standard horizontal (like the xy-plane) or vertical (such as the yz-plane) planes, or they can even be at angles, specified by specific equation constraints, such as \(y = 2\).The transition to using 3D geometry opens up endless possibilities for creating complex and realistic mathematical models that echo those found in real-world scenarios, offering a more accurate depiction of how shapes interact within their surroundings.