In what ratio does the point P(2, −5) divide the segment joining A(−4, 4) and B(6, −11)?

Let P divide AB in the ratio $k$ : 1. Using $x$-coordinates: $\frac{6k+(-4)}{k+1}=2$ → $6k-4=2k+2$ → $4k=6$ → $k=\frac{3}{2}$, so AP : PB = 3 : 2. Check with $y:\frac{1.5(-11)+4}{2.5}=\frac{-12.5}{2.5}=-5$ ✓. Writing the ratio backwards gives 2 : 3. The option 3 : 5 is AP : AB — the part compared with the WHOLE segment (AP = $3\sqrt{13}$, AB = $5\sqrt{13}$) — but the section formula ratio always compares the two PARTS, AP : PB.