Give a geometric description of the set of points whose coordinates satisfy the given conditions.
-$$( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 }

Question

Give a geometric description of the set of points whose coordinates satisfy the given conditions.
-$$( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } < 4$$
A) No set of points satisfy the given relations.
B) All points on the lower hemisphere centered at (10, 2, 2)
C) All points within the lower hemisphere centered at (10, 2, 2)
D) All points outside the lower hemisphere centered at (10, 2, 2)

Quizplus · Accepted Answer

The expression $( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 }$ represents the distance between a point $(x,y,z)$ and the point $(10,2,2)$. Thus, the given inequality states that this distance must be less than 2. This is equivalent to saying that all points within a sphere of radius 2 centered at $(10,2,2)$ satisfy the inequality. However, the inequality only includes the lower hemisphere of this sphere (where $z<2$), and so the answer is C.

Give a Geometric Description of the Set of Points Whose (x−10)2+(y−2)2+(z−2)2<4( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } < 4(x−10)2+(y−2)2+(z−2)2<4

Give a Geometric Description of the Set of Points Whose $( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } < 4$