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# Find the equation of the sphere centered at (-9, 3, 9) with radius 5. Give an equation which describes the intersection of this sphere with the plane z = 10. # Find the equation of the sphere centered at (-9, 3, 9) with radius 5. Give an equation which describes the intersection of this sphere with the plane z = 10.

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Solid Geometry asked 2021-03-02
Find the equation of the sphere centered at (-9, 3, 9) with radius 5. Give an equation which describes the intersection of this sphere with the plane z = 10.

## Answers (1) 2021-03-03
Given: A circle with Center C = (-9,3,9) Radius r=5
General eqation for a sphere with center $$C=(x_0,y_0,z_0)$$ and radius r.
$$(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2$$
Replace $$x_0$$ with -9, $$y_0$$ with 3, $$z_0$$ with 9 and r with 5.
$$(x-(-9))^2+(y-3)^2+(z-9)^2-5^2$$
Simplify:
$$(x+9)^2+(y-3)^2+(z-9)^2=25$$
The intersection of the sphere with the plane z=10 can then be found by replacing z in the equation of the sphere by 10.
$$(x+9)^2+(y-3)^2+(10-9)^2=25$$
Simplify:
$$(x+9)^2+(y-3)^2+1=25$$
Subtract 1 from each side:
$$(x+9)^2+(y-3)^2=24$$
CONCLUSION
Equation sphere: $$(x+9)^2+(y-3)^2+(10-9)^2=25$$
Equation intersection with plane z=10: $$(x+9)^2+(y-3)^2=24$$

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