Q61 / 100 · BITSAT 2024 (Memory Based Paper 2) · Mathematics

Let ABC be a triangle and be $\vec{a}, \vec{b}, \vec{c}$ the position vectors of $\mathrm{A}, \mathrm{B}, \mathrm{C}$ respectively. Let D divide $B C$ in the ratio $3: 1$ internally and $E$ divide AD in the ratio $4: 1$ internally. Let BE meet AC in F . IfE divides BF in the ratio $3: 2$ internally then the position vector of $F$ is

  • A
    $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$
  • B
    $\frac{\vec{a}-2 \vec{b}+3 \vec{c}}{2}$
  • C
    $\frac{\vec{a}+2 \vec{b}+3 \vec{c}}{2}$
  • Correct answer
    D
    $\frac{\vec{a}-\vec{b}+3 \vec{c}}{3}$

Solution

Here we are given that $\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{c}}$ Now P.V of Di.e $\overrightarrow{\mathrm{OD}}=\frac{1 \times \overrightarrow{\mathrm{OB}}+3 \times \overrightarrow{\mathrm{OC}}}{1+3}$ $\begin{aligned} & \overrightarrow{\mathrm{OD}}=\frac{\overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}}{4} \\ & \overrightarrow{\mathrm{OE}}=\frac{4 \overrightarrow{\mathrm{OD}}+\overrightarrow{\mathrm{OA}}}{4+1}=\frac{\frac{4(\overrightarrow{\mathrm{~b}}+3 \overrightarrow{\mathrm{c}})}{4}+\overrightarrow{\mathrm{a}}}{5} \\ & \Rightarrow \overrightarrow{\mathrm{OE}}=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}}{5} \\ & \text { Now, } \overrightarrow{\mathrm{OE}}=\frac{2 \overrightarrow{\mathrm{OB}}+3 \overrightarrow{\mathrm{OF}}}{2+3} \\ & \Rightarrow \overrightarrow{\mathrm{OF}}=\frac{5 \overrightarrow{\mathrm{OE}}-2 \overrightarrow{\mathrm{OB}}}{3} \\ & \Rightarrow \overrightarrow{\mathrm{OF}}=\frac{\frac{5(\vec{a}+\vec{b}+3 \overrightarrow{\mathrm{c}})}{5}-2 \overrightarrow{\mathrm{~b}}}{3} \\ & \Rightarrow \overrightarrow{\mathrm{OF}}=\frac{\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}}{3} \Rightarrow P . V . \text { of } \mathrm{F} \text { is } \frac{\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+3 \overrightarrow{\mathrm{c}}}{3}\end{aligned}$

Hence, the correct answer is option: 'D'.

Relevant Questions