   Chapter 12.5, Problem 61E

Chapter
Section
Textbook Problem

# Find an equation for the plane consisting of all points that are equidistant from the points (1, 0, −2) and (3, 4, 0).

To determine

To find: The equation for the plane that consists of all the points that are equidistant from the points (1,0,2) and (3,4,0).

Explanation

Formula used:

The equation 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:

Consider a point P(x,y,z) in the plane which is equidistant from the points (1,0,2) and (3,4,0).

Let the distance between the points P(x,y,z) and (1,0,2) be d1, and the distance between the points P(x,y,z) and (3,4,0) be d2.

As the distance between P(x,y,z) and (1,0,2) is equal to the distance between the points P(x,y,z) and (3,4,0), write the expression mathematically as follows.

d1=d2 (1)

Substitute x for x1, y for y1, z for z1, 1 for x2, 0 for y2, and 2 for z2 in equation (2),

d1=(1x)2+(0y)2+[(2)z]2

Substitute x for x1, y for y1, z for z1, 3 for x2, 4 for y2, and 0 for z2 in equation (2),

d2=(3x)2+(4y)2+(0z)

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