Recover every valid parallel-plane distance parameter
Find every real value of λ for which both equations describe planes and their perpendicular separation equals the stated positive distance. Establish proportional normals, retain any excluded parameter, then express the distance-equation candidates as ±C√R with C positive rational and R positive square-free. Use R=1 for rational candidates
\[\mathbf n(\lambda)=(\lambda,3,4),\quad e=0,\quad \mathbf r=(x,y,z)\]
\[\Pi_1:\ \mathbf n(\lambda)\cdot\mathbf r=1\]
\[\Pi_2:\ (\lambda-e)\mathbf n(\lambda)\cdot\mathbf r=(\lambda-e)(1+4\lambda)\]
\[\operatorname{distance}(\Pi_1,\Pi_2)=2\]
Report both algebraic candidates through C and R, then mark each candidate valid or invalid for the ORIGINAL pair of equations. A zero normal does not define a plane