For any two points $M$ and $N$ in the $X Y$-plane, let $\overrightarrow{M N}$ denote the vector from

Question

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 $	riangle 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\ $10\frac{	ext { length of the line segment } E F}{	ext { length of the line segment } E S}$ is $\_\_\_\_$ .\

Professor PC Thomas Classes · Answer

(1.2): Let position vectors of $P, Q, R$, $S$ be $\vec{p}, \vec{q}, \vec{r}, \vec{s}$\ $E=\frac{\vec{p}+\vec{r}}{2}, F=\frac{\vec{q}+\vec{r}}{2}$\ $\overrightarrow{S P}+5 \overrightarrow{S Q}+6 \overrightarrow{S R}=0$\ $\Rightarrow \vec{p}-\vec{s}+5(\vec{q}-\vec{s})+6(\vec{r}-\vec{s})=0$\ $\Rightarrow \vec{s}=\frac{\vec{p}+5 \vec{q}+6 \vec{r}}{12}$\ $	herefore \quad|\overrightarrow{E S}|=\left|\frac{5 \vec{q}-5 \vec{p}}{12}ight|$ and $|\overrightarrow{E F}|=\left|\frac{\vec{q}-\vec{p}}{2}ight|$\ $	herefore \frac{|\overrightarrow{E F}|}{|\overrightarrow{E S}|}=\frac{\frac{1}{2}}{\frac{5}{12}}=\frac{6}{5}=1.2$\