Q92 / 150 · MHT CET 2024 (03 May Shift 2) · Mathematics

If $\mathrm{A}(-4,5, \mathrm{P}), \mathrm{B}(3,1,4)$ and $\mathrm{C}(-2,0, \mathrm{q})$ are the vertices of a triangle $A B C$ and $G(r, q, 1)$ is its centroid, then the value of $2 p+q-r$ is equal to

  • Correct answer
    A
    -3
  • B
    -6
  • C
    9
  • D
    4

Solution

$\mathrm{A}(-4,5, \mathrm{p}), \mathrm{B}(3,1,4)$ and $\mathrm{C}(-2,0, \mathrm{q})$ are vertices of triangle $A B C$ $\therefore \quad$ Centroid $\equiv\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{\mathrm{z}_1+\mathrm{z}_2+\mathrm{z}_3}{3}\right)$ $\Rightarrow(\mathrm{r}, \mathrm{q}, 1) \equiv\left(\frac{-4+3-2}{3}, \frac{5+1+0}{3}, \frac{\mathrm{p}+4+\mathrm{q}}{3}\right)$ $\begin{aligned} & \Rightarrow \mathrm{r}=-1, \mathrm{q}=2, \mathrm{p}+\mathrm{q}+4=3 \\ & \Rightarrow \mathrm{r}=-1, \mathrm{q}=2, \mathrm{p}=-3 \\ & \therefore \quad 2 \mathrm{p}+\mathrm{q}-\mathrm{r}=2(-3)+2+1 \\ & =-6+2+1 \\ & =-3 \end{aligned}$

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

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