Position of a Point with Respect to a Straight Line

Any line $ax+by+c=0$ with $c\ne0$ splits the whole plane into two half-planes: the origin side (containing the origin) and the non-origin side. A point $P(x_1,y_1)$ lies on the origin side or the non-origin side according as $ax_1+by_1+c$ and $c$ have the same sign or opposite signs. In particular, if $c>0$, then $P$ is on the origin side when $ax_1+by_1+c$ is positive, and on the non-origin side when it is negative.

This sign idea also gives a quick way to classify the angle between two lines directly from their general-form coefficients, without computing slopes: first rewrite $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ so that both constant terms are positive ($c_1>0,c_2>0$, achieved if necessary by multiplying an equation through by $-1$); then

• if $a_1a_2+b_1b_2<0$, the angle between the lines is acute;
• if $a_1a_2+b_1b_2>0$, the angle between the lines is obtuse. …