For y^2=4ax, compute S1=y1^2-4ax1 at point (x1,y1). S1<0: inside (on the focus side), S1=0: on the parabola, S1>0: outside. Inside means between the two branches in the region containing the focus. The origin (vertex) is on the parabola (S1=0). The focus (a,0) gives S1=-4a^2<0 (inside). Points on the directrix have S1=y1^2+4a^2>0 (always outside). The sign of S1 determines whether tangents from that point exist (S1>0: two tangents, S1=0: one tangent, S1<0: none).