Your idea about how to go about showing it is good.
Write the equation of the line in parametric form, using say the parameter name t.
Now, substitute the formulae(in terms of t) for x, y, and z of that parametric form in for the x, y, and z that appear in the equation for the sphere. That results in a quadratic equation in the parameter t. Now solve that quadratic equation, to get two solutions for t. Plug each of those two values of t into the parametric form of the equation of the line, and those results are the two intersecting points.
Here's how to do those steps in Maple. First, create the points, line, and sphere.
_EnvXName := x: _EnvYName := y: _EnvZName := z:
sphere(s,x^2+y^2+z^2 = 36,[x,y,z]):
Maple can now give the answer right away,
Or you can do each step in Maple,
eL:=Equation(L,'t'); # equation of line
# substitute components of line form into sphere's equation
# solve the quadratic equation
# plug both t values into the line (parametric form).
seq(eval(eL,t=tval),tval in [tsols]);