Q70 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics

If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are position vectors of 4 points such that $2 \vec{a}+3 \vec{b}+5 \vec{c}-10 \vec{d}=\overrightarrow{0}$ then the ratio in which the line joining $\vec{c}$ and $\vec{d}$ divides the line segment joining $\vec{a}$ and $\vec{b}$ is

  • A
    $2: 3$
  • B
    $-1: 2$
  • C
    $2: 1$
  • Correct answer
    D
    $3: 2$

Solution

$\begin{aligned} & 2 \vec{a}+3 \vec{b}+5 \vec{c}-10 \vec{d}=\overrightarrow{0}, \Rightarrow 2 \vec{a}+3 \vec{b}=10 \vec{d}-5 \vec{c} \\ & \Rightarrow \frac{2 \vec{a}+3 \vec{b}}{5}=\frac{2 \vec{d}-\vec{c}}{1} \Rightarrow \frac{2 \vec{a}+3 \vec{b}}{3+2}=\frac{2 \vec{d}-\vec{c}}{2-1} \end{aligned}$ $\therefore$ Line joining $\vec{c}$ and $\vec{d}$ divides the line joining $\vec{a}$ and $\vec{b}$ in 3:2.

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

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