<s>[INST] Equation of the sphere that passes through 4 points Write he equation of the sphere that passes through points
$$a(-5,4,1),b(3,4,-5),c(0,0,4),d(0,0,0)$$
I tried to use four points to draw a geometric shape and then calculate the center of this shape on the basis of the circle that passing  on four points. But I did not succeed

Here is the book answer
  $$x^2+y^2+z^2+54x−58y+4z=0$$

 [/INST] Using the equation for points on spheres:  
$\qquad(x-a)^2+(y-b)^2+(z-c)^2=r^2$
Using coordinates of the four points provided, we have four simultaneous equations to solve for $a, b, c, d$.
\begin{cases}
(-5-a)^2+(4-b)^2+(1-c)^2=r^2  \\
(3-a)^2+(4-b)^2+(-5-c)^2=r^2  \\
(0-a)^2+(0-b)^2+(4-c)^2=r^2  \\
(0-a)^2+(0-b)^2+(0-c)^2=r^2 
\end{cases}
(Substracting third and fourth equation yield $c$, and the first two yield $a$ then $b$, then you have $r$, easy to solve on paper)  
and get a nonnegative solution:
\begin{cases}
a=2  \\
b=\frac{29}{4}  \\
c=2  \\
r=\frac{\sqrt{969}}{4}
\end{cases}
(just solved it, maybe you can check the answers :D )
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