Question 9

Let the position vectors of $P, Q, R, S$ be $\vec{p}, \vec{q}, \vec{r}, \vec{s}$ respectively.

1. From $\vec{SP} + 5 \vec{SQ} + 6 \vec{SR} = \vec{0}$, we have: $(\vec{p} - \vec{s}) + 5(\vec{q} - \vec{s}) + 6(\vec{r} - \vec{s}) = \vec{0}$ $\vec{p} + 5\vec{q} + 6\vec{r} = 12\vec{s} \implies \vec{s} = \frac{\vec{p} + 5\vec{q} + 6\vec{r}}{12}$.
2. $E$ is the midpoint of $PR$, so $\vec{e} = \frac{\vec{p} + \vec{r}}{2}$.
3. $F$ is the midpoint of $QR$, so $\vec{f} = \frac{\vec{q} + \vec{r}}{2}$.
4. Length $EF = |\vec{f} - \vec{e}| = |\frac{\vec{q} + \vec{r}}{2} - \frac{\vec{p} + \vec{r}}{2}| = |\frac{\vec{q} - \vec{p}}{2}| = \frac{1}{2} |\vec{PQ}|$.
5. Vector $\vec{ES} = \vec{s} - \vec{e} = \frac{\vec{p} + 5\vec{q} + 6\vec{r}}{12} - \frac{\vec{p} + \vec{r}}{2} = \frac{\vec{p} + 5\vec{q} + 6\vec{r} - 6\vec{p} - 6\vec{r}}{12} = \frac{5\vec{q} - 5\vec{p}}{12}$.
6. Length $ES = |\vec{ES}| = \frac{5}{12} |\vec{q} - \vec{p}| = \frac{5}{12} |\vec{PQ}|$.
7. The ratio $\frac{EF}{ES} = \frac{(1/2) |\vec{PQ}|}{(5/12) |\vec{PQ}|} = \frac{1}{2} \times \frac{12}{5} = \frac{6}{5} = 1.2$. Final Answer: 1.2