How do you find an equation of the sphere with center (2, -6, 4) and radius 5?

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Answer 1 · 2016-06-16T06:19:16

The equation can be written in the form:

#(x-2)^2+(y-(-6))^2+(z-4)^2 = 5^2#

Given any two points #(x_1, y_1, z_1)# and #(x_2, y_2, z_2)#, the distance between them is given by the formula:

#d = sqrt((x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2)#

So for a sphere with centre #(2, -6, 4)# and radius #5#, every point #(x, y, z)# on the surface satisfies:

#5 = sqrt((x-2)^2+(y-(-6))^2+(z-4)^2)#

Squaring both sides and transposing this becomes:

#(x-2)^2+(y-(-6))^2+(z-4)^2 = 5^2#

This is in the form:

#(x-a)^2+(y-b)^2+(z-c)^2 = r^2#

with #(a, b, c) = (2, -6, 4)# and #r=5#.

Note the similarity with the equation of a circle with centre #(h, k)# and radius #r#, namely:

#(x-h)^2+(y-k)^2 = r^2#