Family of Lines

A family of lines is the set of all lines satisfying one common stated condition. Even though the equation of a straight line $ax+by+c=0$ shows three symbols $a,b,c$, dividing through by whichever of $a,b$ is nonzero shows the equation genuinely has only two independent arbitrary constants (e.g. rewritten as $Ax+y+C=0$, or as $y=mx+b$) — so a straight line's equation always needs exactly two independent conditions to pin it down uniquely, never more, never fewer.

Imposing just one linear relation among those two constants leaves a single free constant, called the parameter; the resulting infinite collection of lines (one for each value of the parameter) is called a one-parameter family of lines, and the free constant is the parameter of the family. Leaving both constants free (subject only to the original single condition defining a line) gives a two-parameter family. …