Putnam 1959 B5

Find the equation of the smallest sphere which is tangent to both of the lines $$\begin{pmatrix} xyz \end{pmatrix} =\begin{pmatrix} t+1 2t+4 -3t +5 \end{pmatrix},\;\;\;\begin{pmatrix} xyz \end{pmatrix} =\begin{pmatrix} 4t-12 -t+8 t+17 \end{pmatrix}.$$