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.

Question
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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