JEE Advanced 2025 — Paper 1 — Question 23

$\because \overrightarrow{\mathrm{SP}}+5 \overrightarrow{\mathrm{SQ}}+6 \overrightarrow{\mathrm{SR}}=\overrightarrow{0}$

(Let P.V. of P be $\overrightarrow{\mathrm{p}}, \mathrm{Q}$ be $\overrightarrow{\mathrm{q}}$ and R be $\overrightarrow{\mathrm{r}}$ )

$\Rightarrow(\overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{s}})+5(\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{s}})+6(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{s}})=\overrightarrow{0}$

$\Rightarrow \vec{S}=\frac{\vec{p}+5 \vec{q}+6 \vec{r}}{12}$

$\Rightarrow \overrightarrow{\mathrm{EF}}=\left(\frac{\overrightarrow{\mathrm{q}}-\overrightarrow{\mathrm{p}}}{2}\right)$

$\Rightarrow \overrightarrow{E S}=\left(\frac{5 \vec{q}-5 \vec{p}}{12}\right)=\frac{5}{12}(\vec{q}-\vec{p})$

$\therefore \frac{|\overrightarrow{\mathrm{EF}}|}{|\overrightarrow{\mathrm{ES}}|}=\frac{6}{5}=1.2$