Subject: math
Name: sasha

could a "Family of Lines" have both parallel and intersecting equations that aren't related directly but otherwise related by each other to create a curve

Sasha,

You might have to clarify the question. Does the family of all tangents to a circle qualify as a family you might be interested in? Each tangent has one parallel tangent, and the rest intersect it. For example, the family of tangents to the unit circle x² + y² = 1 have the form ax + by = 1, for numbers a, b that satisfy a² + b² = 1.

Chris