   Chapter 12.5, Problem 62E

Chapter
Section
Textbook Problem

# Find an equation for the plane consisting of all points that equidistant from the points (2, 5, 5) and (−6, 3, 1).

To determine

To find: The equation for the plane that consists of all points that are equidistant from the points (2,5,5) and (6,3,1) .

Explanation

Consider a point P(x,y,z) in the plane which is equidistant from the points (2,5,5) and (6,3,1) .

Consider the distance between the points P(x,y,z) and (2,5,5) is d1 , and the distance between the points P(x,y,z) and (6,3,1) is d2 .

As the distance between P(x,y,z) and (2,5,5) is equal to the distance between the points P(x,y,z) and (6,3,1) , write the expression mathematically as follows.

d1=d2 (1)

Formula used:

Write the expression to find the distance between two points Q(x1,y1,z1) and R(x2,y2,z2) .

|QR|=(x2x1)2+(y2y1)2+(z2z1)2 (2)

Calculation distance between the points P(x,y,z) and (2,5,5) :

Substitute x for x1 , y for y1 , z for z1 , 2 for x2 , 5 for y2 , and 5 for z2 in equation (2),

d1=(2x)2+(5y)2+(5z)2

Calculation distance between the points P(x,y,z) and (6,3,1) :

Substitute x for x1 , y for y1 , z for z1 , 6 for x2 , 3 for y2 , and 1 for z2 in equation (2),

d2=[(6)x]2+(3y)2+(1z)2

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