Does the point Q(3, 0) lie on the segment joining A(−3, 6) and B(6, −3), and if so, in what ratio does it divide the segment?

Suppose Q divides AB in the ratio $k$ : 1. Q lies on the $x$-axis, so its $y$-coordinate must be 0: $\frac{k(-3)+1(6)}{k+1}=0$ → $-3k+6=0$ → $k=2$, i.e. the ratio 2 : 1. Now verify the $x$-coordinate with that ratio: $\frac{2(6)+1(-3)}{3}=\frac{9}{3}=3$ ✓, matching Q(3, 0), so Q genuinely lies on AB. Using the ratio the wrong way round, 1 : 2, would place the point at $(\frac{6+2(-3)}{3},\frac{-3+12}{3})=(0,3)$ — on the $y$-axis, not at Q.