a) The sphere has radius equal to the distance between the center and the given point:
[tex]\sqrt{(6+1)^2+(-2-2)^2+(3-1)^2}=\sqrt{69}[/tex]
So the sphere has equation
[tex](x-6)^2+(y+2)^2+(z-3)^2=69[/tex]
b) The sphere intersects the [tex]y[/tex]-[tex]z[/tex] plane whenever [tex]x=0[/tex]:
[tex](-6)^2+(y+2)^2+(z-3)^2=69\implies(y+2)^2+(z-3)^2=33[/tex]
which is the equation of the circle centered at (-2, 3) with radius [tex]\sqrt{33}[/tex].
c) Complete the squares:
[tex]x^2+y^2+z^2-8x+2y+6z+1=0[/tex]
[tex](x^2-8x+16)+(y^2+2y+1)+(z^2+6z+9)+1=16+1+9[/tex]
[tex](x-4)^2+(y+1)^2+(z+3)^2=25[/tex]
So this sphere has radius 5 and is centered at (4, -1, -3).
