Determining an equation for a sphere

Question

my question is:

Given a line S: $$ \left\{ \begin{array}{c} x+y-1=0 \\ y+3z-2=0 \\ \end{array} \right. $$ I need to determine the equation of a sphere having the center on line S and tangent to the plane $z=0.$ The cartesian equation of the line S $= ( -1,2,0 ) + ( 3,-3,1 )t.$

Thanks in advance.

Answer 1

Take any point on the line and consider the sphere centered on this point, with a radius equal to $|z|$. It fulfills the requirements.

Answer 2

Hint

A point of $S$ has cordinates $$y=2-3t$$ $$x=1-y=3t-1$$ $$z=t$$

its distance to plane $z=0$ is the radius

$$r=\frac{|t|}{1}$$

the equation of the sphere is

$$(x+1-3t)^2+(y+3t-2)^2+(z-t)^2=t^2$$