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 (2,3,6)(2, -3, 6) and touch the xyxy-plane, we follow these steps:

1. Understanding the Problem:

  • The center of the sphere is given by the point (2,3,6)(2, -3, 6).
  • The sphere touches the xyxy-plane, meaning the distance from the center of the sphere to the xyxy-plane is equal to the radius rr of the sphere.

2. Distance from the Center to the xyxy-Plane:

  • The xyxy-plane is characterized by the equation z=0z = 0.
  • The distance from the point (2,3,6)(2, -3, 6) to the xyxy-plane is simply the absolute value of the zz-coordinate of the center. Distance=6=6\text{Distance} = |6| = 6
  • Therefore, the radius rr of the sphere is 66.

3. Equation of the Sphere:

  • The general equation of a sphere with center (h,k,l)(h, k, l) and radius rr is given by: (xh)2+(yk)2+(zl)2=r2(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
  • Substituting h=2h = 2, k=3k = -3, l=6l = 6, and r=6r = 6, the equation becomes: (x2)2+(y+3)2+(z6)2=62(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 6^2 (x2)2+(y+3)2+(z6)2=36(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 36

Conclusion:

The equation of the sphere is: (x2)2+(y+3)2+(z6)2=36(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 36

This sphere has its center at (2,3,6)(2, -3, 6) and touches the xyxy-plane.


Would you like more details on any part of this solution?

Here are some related questions to explore:

  1. What is the equation of a sphere centered at (h,k,l)(h, k, l) with a different radius rr?
  2. How would the equation change if the sphere touched the yzyz-plane instead?
  3. How can you determine if a given point lies on this sphere?
  4. What is the distance between two points in 3-dimensional space?
  5. 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

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Suitable Grade Level

Grades 10-12