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.

Solution:

Given points A(-1, 5, 3) and B(6, 2, -2)

Let P be the point equidistant from A and B such that PA = PB

We know that distance formula between two 3 dimensional points

d= √(x - x’)² + (y - y’)² + (z - z’)²

PA = √(x + 1)² + (y - 5)² + (z - 3)²

PB = √(x - 6)² + (y - 2)² + (z + 2)²

Now, PA = PB

√(x + 1)² + (y - 5)² + (z - 3)² = √(x - 6)² + (y - 2)² + (z + 2)²

Squaring on both sides

(x + 1)² + (y - 5)² + (z - 3)² = (x - 6)² + (y - 2)² + (z + 2)²

x² + 2x + 1 + y² - 10y + 25 + z² - 6z + 9 = x² - 12x + 36 + y² - 4y + 4 + z² + 4z + 4

By simplifying we get 14x - 6y - 10z = 9

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.

Summary:

An equation of the set of all points equidistant from the points A(-1, 5, 3) and B(6, 2, -2) is 14x - 6y - 10z = 9.