The familiar quadratic curve can be defined entirely by equal distances.
Use focus (0,p) and directrix y=−p. Equal distances give √[x²+(y−p)²]=y+p on the parabola. Squaring and simplifying yields x²=4py. Conversely, substituting the quadratic verifies both distances, since y+p is positive.
Distance to a line means perpendicular distance, not distance to an arbitrary point on the line.