Spheres

# Spheres - Math 17C Kouba Three-Dimensional(3-D Space RECALL...

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Unformatted text preview: Math 17C Kouba Three-Dimensional (3-D) Space RECALL : Consider two points (x 1, y 1) and (x 2, y 2) in two-dimensional space. The midpoint of the line segment joining these two points is given by (x1+x2:Y1+Y2) (x1,y1) 2 2 0 "\ ThembeMeen (x,,y1) (xz.y2) these two points is ' ~ . D=¢(X2~X1‘)2+(Y2-y1)2. “" (X2»Y2) RECALL : The set of all points (x, y) in two-dimensional space which are a distance r from a fixed point (h, k) is a circle (with center (h, k) and radius r) given by the equation (x-h)2+(y-k)2 = r2. Let (x 1, y 1, z 1) and (x 2. y 2, z 2) be two points in three-dimensional space. The midpoint of the line segment joining these two points is given by (XI+XZIY1+y2,Zi+22> (x,,y1,z1) 2 2 2 ’ .‘\\ [ ““\ Thesﬂstamebetween (Xvi/1J1) “- (xz.y2.22) these two points is ’x o ‘\ [him-x,)2+(y2-y,)2+(22—z,)2 ‘ DEFINITION : The set of all points (x, y, z) in three-dimensional space which are a distance r from a fixed point (h, k, I) is a sphere (with center (h, k, l) and radius r given by the equation (x--h)2+(y-k)2+(z-l)2 = r2. Example : Find the center and radius of each of the following spheres. 1. 2x2+2y2+222 = 32 center (0, 0. 0) radius4 2. x2+y2+22—4x+6y =17 center (2,—3, O) radius «93 Example : The diameter of a sphere h Determine an equation for this sphere. as endpoints (1, 3, 0) and (-2, 4, 6) . (x+1/2)2+(y--7/2)2+(z--3)2 = 23/2 Example : Find and simplify an equation for all points (x, y, z) in three-dimensional space which are equidistant from the point (1 ,-2, 3) and the plane 2 =-1. z=1/a(x—1)2+1/8(y+2)2+1 ...
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Spheres - Math 17C Kouba Three-Dimensional(3-D Space RECALL...

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