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:

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

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