For any two points $M$ and $N$ in the $X Y$-plane, let $\overrightarrow{M N}$ denote the vector from $M$ to $N$, and $\overrightarrow{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $X Y$-plane. Let $S$ be a point inside the triangle $\triangle P Q R$ such that $\overrightarrow{S P}+5 \overrightarrow{S Q}+6 \overrightarrow{S R}=\overrightarrow{0}$. Let $E$ and $F$ be the mid-points of the sides $P R$ and $Q R$, respectively. Then the value of $\frac{\text { length of the line segment } E F}{\text { length of the line segment } E S}$ is ________

Solution

$\begin{aligned} &\because \overrightarrow{\mathrm{SP}}+5 \overrightarrow{\mathrm{SQ}}+6 \overrightarrow{\mathrm{SP}}=\overrightarrow{0}\\ &\text { (Let P.V. of } P \text { be } \vec{p}, Q \text { be } \vec{q} \text { and } R \text { be } \vec{r} \text { ) }\\ &\begin{aligned} & \Rightarrow(\vec{p}-\vec{s})+5(\vec{q}-\vec{s})+6(\vec{r}-\vec{s})=\overrightarrow{0} \\ & \Rightarrow \vec{S}=\frac{\vec{p}+5 \vec{q}+6 \vec{r}}{12} \\ & \Rightarrow \overrightarrow{E F}=\left(\frac{\vec{q}-\vec{p}}{2}\right) \\ & \Rightarrow \overrightarrow{E S}=\left(\frac{5 \vec{q}-5 \vec{p}}{12}\right)=\frac{5}{12}(\vec{q}-\vec{p}) \\ & \therefore \frac{|\overrightarrow{E F}|}{|\overrightarrow{E S}|}=\frac{6}{5}=1.2 \end{aligned} \end{aligned}$

Hence, the correct answer is 1.2.