Q72 / 160 · TS EAMCET 2023 (14 May Shift 1) · Mathematics

If $\mathrm{A}(1,2,3), \mathrm{B}(3,7,-2), \mathrm{C}(6,7,7)$ and $\mathrm{D}(-1,0,-1)$ are points in a plane, then the vector equation of the line passing through the centroids of $\triangle \mathrm{ABD}$ and $\triangle \mathrm{ACD}$ is

  • A
    $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}})+\mathrm{t}(\hat{\mathrm{j}}+4 \hat{\mathrm{k}})$
  • Correct answer
    B
    $\overrightarrow{\mathrm{r}}=(1+t) \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+3 \mathrm{t} \hat{\mathrm{k}}$
  • C
    $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\mathrm{t}(\hat{\mathrm{i}}+3 \hat{\mathrm{j}})$
  • D
    $\vec{r}=(\hat{i}+\hat{j}+\hat{k})+t(2 \hat{i}-\hat{j})$

Solution

$\mathrm{A}(1,2,3) ; \mathrm{B}(3,7,-2) ; \mathrm{C}(6,7,7) ; \mathrm{D}(-1,0,-1)$ Centroid of $\triangle \mathrm{ABD}=(1,3,0)$ Centroid of $\triangle \mathrm{ACD}=(2,3,3)$ Equation of vector passing through $\vec{a}$ and $\vec{b}$ $\begin{aligned} & \vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a}) \\ & \vec{r}=(\hat{i}+3 \hat{j})+\lambda(\hat{i}+3 \hat{k}) \\ & \vec{r}=(1+\lambda) \hat{i}+3 \hat{j}+3 \lambda \hat{k} \text { or } \vec{r}=(1+t) \hat{i}+3 \hat{j}+3 t \hat{k} \end{aligned}$

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

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