The family of lines through the intersection of L1: a1x + b1y + c1 = 0 and L2: a2x + b2y + c2 = 0 is L1 + lambda*L2 = 0. This one-parameter family includes every line through the intersection point except L2 itself (which corresponds to lambda approaching infinity). The strategy is: write the family equation, apply the additional constraint (passes through a point, has a given slope, is tangent to a curve, is at a given distance from a point), solve for lambda, and substitute back. This approach is vastly more efficient than finding the intersection point first and then constructing the required line. It is especially powerful when the intersection coordinates are irrational or fractional, as the lambda parameter keeps calculations clean.