Necessary & Sufficient condition for the line $ax+by+c=0$ to pass through the 1st quadrant

Question

What is the necessary and sufficient condition for the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, to pass through the first quadrant?

I could find the points at which the line crosses the coordinate axes, namely $(\frac{-c}{a},0)$ and $(0,\frac{-c}{b})$. But I guess I have to relate the problem to the slope of the line.

Answer 1

Considering four cases

• $-c/a\lt 0$ and $-c/b\lt 0$

• $-c/a\lt 0$ and $-c/b\gt 0$

• $-c/a\gt 0$ and $-c/b\lt 0$

• $-c/a\gt 0$ and $-c/b\gt 0$

gives that the necessary and sufficient condition is $$-\frac ca\gt 0\quad\text{or}\quad -\frac cb\gt 0,$$ i.e. $$\color{red}{ac\lt 0\quad\text{or}\quad bc\lt 0}$$