Rational curves such as rational Bézier curves and NURBS are widely being used in CAD and CAGD, and defining them as homogeneous curves in homogeneous space has many advantages in geometric processing. In this paper, some properties of quadratic rational Bézier curves and NURBS defined in homogeneous space are discussed. First, the relationship between the two kinds of homogeneous curves is studied. Secondly, the implicit form of quadratic homogeneous Bézier curves is derived, and by using the implicit form, the inclusion test and the conic classification (hyperbola, parabola, and ellipse) are performed. Finally, the fact that a whole conic can be represented as one homogenized NURBS is shown. The properties discussed here are more general than those of conventional rational curves.