   Chapter 12.1, Problem 45E

Chapter
Section
Textbook Problem

# Find an equation of the set of all points equidistant from the points A(−1, 5, 3) and B(6, 2, −2). Describe the set.

To determine

To find: An equation of the set of all points equidistant from the points A(1,5,3) and B(6,2,2) , determine the set of all points equidistant from the points A(1,5,3) and B(6,2,2) .

Explanation

Consider a point P(x,y,z) .

The distance between the points P and A is equals to the distance between the points P and B.

Formula:

Write the expression to find the distance between two points P1(x1,y1,z1) and P2(x2,y2,z2) .

|P1P2|=(x2x1)2+(y2y1)2+(z2z1)2 (1)

Calculation distance from point P(x,y,z) to the point A(1,5,3) :

In equation (1), substitute x for x1 , y for y1 , z for z1 , 1 for x2 , 5 for y2 , and 3 for z2 .

|PA|=(1x)2+(5y)2+(3z)2=[(1+x)]2+(5y)2+(3z)2=(1+x)2+(5y)2+(3z)2

Calculation distance from point P(x,y,z) to the point B(6,2,2) :

In equation (1), substitute x for x1 , y for y1 , z for z1 , 6 for x2 , 2 for y2 , and 2 for z2 .

|PB|=(6x)2+(2y)2+(2z)2=(6x)2+(2y)2+[(2+z)]2=(6x)2+(2y)2+(2+z)2

As the distance |PA| is equals to the distance |PB| , write the equation as follows

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