Family of straight lines

A family of intersecting straight lines, is the set of infinite straight lines that pass through a point P(p₁,p₂):

Example: y – 3 = m (x – 2), m Є R, is the family of lines that pass through the point (2,3).

A family of parallel straight lines, is the set of infinite straight lines that are parallel to a line r : Ax + By + C = 0:

Example: 3x + 2y + k = 0, k€R, is the family of lines that are parallel to 3x + 2y – 15 = 0

Exercises:

1.- Find the equation of the line that passes through the intersection point of lines r: 3x + y - 1 = 0 and s: x - y + 5 = 0 and is parallel to x + y = 0.

2.- Find the equation of the line that passes through the points (3,3) and (-1,-1), by using the family of intersecting straight lines of one of them.

Solutions: a) x + y -3 = 0; b) y - 3 = 1·(x - 3)