What is the intersection of this sphere with the YZ-plane? This intriguing question delves into the realm of geometry, where we explore the intersection of a three-dimensional sphere with a two-dimensional plane. As we embark on this journey, we will uncover the equations, techniques, and applications associated with this fascinating topic.
To begin our exploration, we must first establish a clear understanding of the sphere and the YZ-plane. A sphere is a three-dimensional shape defined by a set of points equidistant from a central point. The YZ-plane, on the other hand, is a two-dimensional plane that lies perpendicular to the X-axis and contains both the Y-axis and the Z-axis.
1. Defining the Sphere and YZ-Plane
A sphere is a three-dimensional surface that is equidistant from a fixed point called the center. The equation of a sphere in 3D space is given by:
(x
- h)² + (y
- k)² + (z
- l)² = r²
where (h, k, l) is the center of the sphere and r is the radius.
The YZ-plane is a plane that passes through the Y-axis and the Z-axis. The equation of the YZ-plane is given by:
x = 0
The YZ-plane divides the 3D space into two half-spaces.
2. Determining the Intersection: What Is The Intersection Of This Sphere With The Yz-plane
The intersection of a sphere and a plane is a circle or an ellipse. The equation for finding the intersection of a sphere and a plane is given by:
(x
- h)² + (y
- k)² + (z
- l)² = r²
ax + by + cz + d = 0
where (h, k, l) is the center of the sphere, r is the radius, and (a, b, c, d) are the coefficients of the plane equation.
To find the intersection of the given sphere and YZ-plane, we can substitute x = 0 into the sphere equation:
(0
- h)² + (y
- k)² + (z
- l)² = r²
y² + (z
l)² = r²
This is the equation of a circle in the YZ-plane.
3. Analyzing the Result
The resulting point(s) of intersection represent the points where the sphere and the YZ-plane meet. In this case, the intersection is a circle. The circle lies in the YZ-plane and its center is (0, k, l).
The radius of the circle is r, which is the radius of the sphere.
4. Applications and Examples
Finding the intersection of a sphere and the YZ-plane has applications in various fields, such as:
- Computer graphics: To determine the visibility of objects in a 3D scene.
- Robotics: To calculate the trajectory of a robot arm.
- Architecture: To design curved surfaces and structures.
Here is an example of how to find the intersection of a sphere and the YZ-plane in a real-world application:
| Problem Statement | Solution | Application |
|---|---|---|
| A spherical water tank with radius 5 meters is filled with water. The water level is 3 meters high. Find the area of the water’s surface. | The equation of the sphere is (x
|
Calculating the surface area of a liquid in a spherical container. |
FAQ Guide
What is the equation of a sphere?
The equation of a sphere with center (h, k, l) and radius r is given by (x – h)^2 + (y – k)^2 + (z – l)^2 = r^2.
What is the equation of the YZ-plane?
The equation of the YZ-plane is x = 0.
How do I find the intersection of a sphere and a plane?
To find the intersection of a sphere and a plane, we substitute the equation of the plane into the equation of the sphere and solve for the coordinates of the intersection points.
