How do I find the equation of a sphere that passes through the origin and whose center is #(4, 1, 2)#?

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Answer 1 · 2015-10-28T21:05:52

The equation is: #(x-4)^2+(y-1)^2+(z-2)^2=21#

The general equation of a sphere with a center #C=(x_c,y_c,z_c)# and radius #r# is:

#(x-x_c)^2+(y-y_c)^2+(z-z_c)^2=r^2#

In this case center is given #C=(4,1,2)#.

To calculate the radius we use the second point given - the origin. So the radius is the distance between point #C# and the origin:

#r=sqrt((x_C-x_O)^2+(y_C-y_O)^2+(z_C-z_O)^2)=#

#=sqrt((4-0)^2+(1-0)^2+(2-0)^2)=sqrt(16+1+4)=sqrt(21)#

Now when we have all required data we can write the equation of the sphere:

#(x-4)^2+(y-1)^2+(z-2)^2=21#