Obtain the equation of the sphere which passes through the points $(1,0,0),(0,1,0),(0,0,1)$ and has its radius as small as possible. Obtain the equation of the sphere which passes through the points $(1,0,0),(0,1,0),(0,0,1)$ and has its radius as small as possible.



Let  the sphere passes through $(x_1,y_1,z_1)$
Then i obtained the equation of sphere as $(x^2+y^2+z^2)(x_1+y_1+z_1)-(x_1^2+y_1^2+z_1^2-1)(x+y+z)+x_1^2+y_1^2+z_1^2-x_1-y_1-z_1=0$
I am stuck here.I do not know how to minimize the radius.The answer given in my book is $3(x^2+y^2+z^2)-2(x+y+z)-1=0$
If the center of the sphere is $(a,b,c)$, then the square of the distance from $(a,b,c)$ to each of the points $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ is the same. Solve these equations to get a simple relation among $a$, $b$, and $c$. Then minimize the distance from the center to any one of the points.
Let the equation of sphere is  $x^2+y^2+z^2+2ux+2vy+2wz+d=0$
It's given that the sphere passes through the points $(1,0,0),(0,1,0),(0,0,1)$
Therefore we get $u=v=w=-1/2(d+1)$
We know that $r^2=u^2+v^2+w^2-d$
                     Put the values of $u, v,w$:
                    $r^2=3/4((d+1/3)^2+8/4)$
We have to make $r$ as small as possible 
Therefore $r$  will be smallest if $d=-1/3$ hence put the values of u, v, W, and d in equation of sphere and solve it.
We get the required result