Find the equation that describes the set of points (x, y, z)...?

Question

that are 5 units from the point (2, -1, 6). I know how to find -a- point that is 5 units away, but I have no idea about an equation that includes all points 5 units away. Please help!

Answer 1

Using the given point (2, -1, 6) and a general point (x, y, z), substitute into the distance formula (which is just the Pythagorean Theorem extended to 3D):

dist. = √[(difference in x)^2 + (difference in y)^2 + (difference in z)^2]

So,

5 = √[(x - 2)^2 + (y + 1)^2 + (z - 6)^2]

or, squaring both sides,

25 = (x - 2)^2 + (y + 1)^2 + (z - 6)^2

This equation makes sense when considering a similar problem in 2D:

The equation that describes the set of points (x, y) that are 5 units from the point (2, -1) would be the equation of a circle with radius 5, or (x - 2)^2 + (y + 1)^2 = 25