Math Problem Statement
Find equations of the spheres with center (2,-3,6) that touch the xy-plane
Solution
To find the equations of the spheres that have their center at and touch the -plane, we follow these steps:
1. Understanding the Problem:
- The center of the sphere is given by the point .
- The sphere touches the -plane, meaning the distance from the center of the sphere to the -plane is equal to the radius of the sphere.
2. Distance from the Center to the -Plane:
- The -plane is characterized by the equation .
- The distance from the point to the -plane is simply the absolute value of the -coordinate of the center.
- Therefore, the radius of the sphere is .
3. Equation of the Sphere:
- The general equation of a sphere with center and radius is given by:
- Substituting , , , and , the equation becomes:
Conclusion:
The equation of the sphere is:
This sphere has its center at and touches the -plane.
Would you like more details on any part of this solution?
Here are some related questions to explore:
- What is the equation of a sphere centered at with a different radius ?
- How would the equation change if the sphere touched the -plane instead?
- How can you determine if a given point lies on this sphere?
- What is the distance between two points in 3-dimensional space?
- How can we find the equation of a sphere that passes through a specific point?
Tip: Always remember that the radius of a sphere touching a plane is the perpendicular distance from the center to that plane.
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Math Problem Analysis
Mathematical Concepts
Geometry
Equation of a Sphere
Coordinate Geometry
Formulas
Equation of a Sphere
Theorems
-
Suitable Grade Level
Grades 10-12
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